There is shari'ah and there is fiqh. There's tremendous overlap between the two, but it's possible think that shari'ah is the higher principles and the fiqh is the actual detailed rules. Sort of. The popular usage can be quite confusing: in opinion polling, in Muslim-majority countries, many more people want "shari'ah" than want a system based entirely on fiqh. Look at the data for this 2013 survey. It's a recent survey, but it's a pattern that holds up before this, as well--within the Ottoman Empire, kanun and örf were important non-Quranic sources of law.

You don't have to read it all, you can just look at the linked tables. The third table shows massive support for shari'a among Muslims in the Middle East and North Africa, South Asia, Southeast Asia, and Sub-Saharan Africa (weaker support among the Turkic/Central Asian countries, and European Muslims in the Balkans and Russia). The following charts don't represent the whole Muslim population in these countries, but only for the subset of the population that supports shari'ah, ranging from 8% in Azerbaijan to 99% in Afghanistan. Notice that there's high levels of support for deciding matters of family law with religious law (inheritance, divorce, etc). There's a big range. Of people who support 'shari'ah', 24% of people in Bosnia and 95% of people in Egypt want religious judges deciding matters of family law. Notice two things: 1) Bosnia and Kosovo not withstanding, among shari'ah supporters, there's a fairly high support among shari'ah supporters for having religious judges decide family law cases. But 2) even here, in the least controversial category, a notable portion of "shariah supporters" don't letting religious law rule in practice

Notice also that support weakens even more when it comes to actually implementing the normal punishments prescribed in fiqh. The Qur'anic punishment for theft is pretty unambiguous:

As to the thief, male or female, cut off his or her hands: a punishment by way of example, from Allah, for their crime: and Allah is Exalted in power. (Qur'an 5:38)

Yet "corporal punishment for things like theft" has much weaker support among those who say they support shari'ah than using religious law to decide family matters does. In many countries, less than half of people who support "shari'ah" support the prescribed fiqh punishment for crimes like theft. Other issues commonly assumed to be necessarily part of shari'ah, like stoning as punishment for adultery and the death penalty for apostasy, have even weaker support. While those are both also explicitly set out in the Qur'an and pretty much universally agreed upon by traditional Muslim scholarship, in many places most people who support shari'ah do not support these Qur'anic punishments. So shari'ah is, for many people across the Muslim world, not reducible to merely the rules of fiqh. It's something more.

But first, two brief case studies: In Indonesia, 72% of people are in favor making Shari'a the law of the land. That seems high, I'm sure. Among that 72% of people who say Shari'ah should be law of the land, 71% say shari'ah should be used in family law. But only 45% of those who want Shari'ah want to reintroduce corporal punishment for things like theft. Only 48% want stoning of adulterers. Only 18% want the death penalty for apostates. This is a fairly typical pattern. "Shari'ah" as a concept has high support, and there's a general but not universal consensus that this means for family law, but there's an aversion among many "shari'ah supporters" to institute the fiqh punishments.

In other, more conservative counties (in this survey, Egypt, Afghanistan, Pakistan, Palestine) that same patterns doesn't apply. In Egypt, 74% of Muslims want shari'ah. Of those, 95% want it in family courts, 70% corporal punishment for theft, 81% stoning for adultery, 86% capital punishment for apostasy. Here, shari'ah and fiqh mean pretty close to the same thing for people (though even here, not universally overlapping).

What gives? In the most conservative countries, shari'ah means roughly want you've heard about, with people understanding shari'ah as implementing all the hudud punishments laid out in the Qur'an and Hadith, and worked out in detail by generations of Muslim scholars. In places like Egypt, shari'ah means to most people that specific system of jurisprudence. These rulings are very fined grained, but can vary slightly between the five or so major madhhabs (schools of juridical thought--think of how we have common law in the U.S., U.K., and Australia, but they all differ slightly, but also imagine these countries can't make new laws, only rule on the original body of common law), but on the examples given, there's pretty much universal consensus among the schools (these three examples are all hudud punishments, which is the class of punishments specifically described in the Qur'an or, more rarely, in Hadith; compare to tazir, qisas, and diyya, which are the other important areas of fiqh that less fixed by direct Qur'anic dictum). However, most of the people who "support implementing shari'ah", especially those from outside those conservative countries, don't actually support what implementing what we think of in the West as shari'ah punishments. Shari'ah and fiqh are clearly different for many people.

So if shari'ah doesn't just mean killing apostates and cutting off the hands of thieves, then what does it mean? In all places, shari'ah is a sense of justice. I think the best explanation is in Noah Feldman's article "Why Shariah?". It's published in the New York Times Magazine, not an academic journal, so it's actually a good read (Feldman himself is a law professor and historian, so it's also accurate). But in case you don't read the whole thing, here's one of the important sections of the article:

One reason for the divergence between Western and Muslim views of Shariah is that we are not all using the word to mean the same thing. Although it is commonplace to use the word “Shariah” and the phrase “Islamic law” interchangeably, this prosaic English translation does not capture the full set of associations that the term “Shariah” conjures for the believer. Shariah, properly understood, is not just a set of legal rules. To believing Muslims, it is something deeper and higher, infused with moral and metaphysical purpose. At its core, Shariah represents the idea that all human beings — and all human governments — are subject to justice under the law.

In fact, “Shariah” is not the word traditionally used in Arabic to refer to the processes of Islamic legal reasoning or the rulings produced through it: that word is fiqh, meaning something like Islamic jurisprudence. The word “Shariah” connotes a connection to the divine, a set of unchanging beliefs and principles that order life in accordance with God’s will. Westerners typically imagine that Shariah advocates simply want to use the Koran as their legal code. But the reality is much more complicated. Islamist politicians tend to be very vague about exactly what it would mean for Shariah to be the source for the law of the land — and with good reason, because just adopting such a principle would not determine how the legal system would actually operate.

Shariah is best understood as a kind of higher law, albeit one that includes some specific, worldly commands. All Muslims would agree, for example, that it prohibits lending money at interest — though not investments in which risks and returns are shared; and the ban on Muslims drinking alcohol is an example of an unequivocal ritual prohibition, even for liberal interpreters of the faith. Some rules associated with Shariah are undoubtedly old-fashioned and harsh. Men and women are treated unequally, for example, by making it hard for women to initiate divorce without forfeiting alimony. The prohibition on sodomy, though historically often unenforced, makes recognition of same-sex relationships difficult to contemplate. But Shariah also prohibits bribery or special favors in court. It demands equal treatment for rich and poor. It condemns the vigilante-style honor killings that still occur in some Middle Eastern countries. And it protects everyone’s property — including women’s — from being taken from them. Unlike in Iran, where wearing a head scarf is legally mandated and enforced by special religious police, the Islamist view in most other Muslim countries is that the head scarf is one way of implementing the religious duty to dress modestly — a desirable social norm, not an enforceable legal rule. And mandating capital punishment for apostasy is not on the agenda of most elected Islamists. For many Muslims today, living in corrupt autocracies, the call for Shariah is not a call for sexism, obscurantism or savage punishment but for an Islamic version of what the West considers its most prized principle of political justice: the rule of law.

So your question, "What exactly is Sharia Law?", is impossible to answer fully because shari'ah clearly means different things to different people. Sometimes, it's a specific set of laws and punishments. Often, however, it's a general principle of equal and divine justice carrying with it only some of those specific laws and perhaps none of the punishments.

I'm actually going to disagree with /u/yodatsracist on a few points. In discussing Egypt he seems to suggest that implementing the Sharia is part and parcel with implementing the hudud in the popular imagination. Article 2 of the 1971 Egyptian constitution (as amended in 1980) says that "Islam is the religion of the state and Arabic its official language. Principles of Islamic law (Shari’a) are the principal source of legislation." Egypt does not enforce the hudud. Enforcement of the Sharia in that strictly narrow sense has not been a major part of the political agenda of Egypt's Islamist parties, who have instead emphasized things like the Islamic-ness of Egypt as defined by the pre-existing constitution and pushed to have Egypt's clerics and religious scholars (i.e. al-Azhar) take a more direct role in supervising the legislation of the state.

I also don't think the Sharia is quite as inscrutable as he's suggesting. Sharia is the religious law of Islam. Fiqh is the jurisprudence that interprets and applies that law. Etymologically Sharia refers to a path or way, in that it's similar to the word "Sunna" which also refers to a path. It shares a linguistic "root" with the commonplace Arabic words for legislation and legitimacy. The sources of the Sharia for Sunnis include, in order of preference, the Quran, the prophetic hadith recording the declarations of the prophet and emulations of his example, as well as, to a lesser extent, emulation of the salaf and, in the Maliki school, emulation of the customary practices of the people of Medina. Fiqh takes from these primary sources an interpretative method, developed in large part by al-Shafii and adds, in addition to Quran and Sunna and in order of preference, the consensus of the Islamic community, analogical reasoning and individual reasoning.

I don't think the Feldman article is describing some higher sense of justice so much as the fact that the sharia, for the vast majority of the world's muslims, describes a personal religious law which in many senses defines ones own personal behavior as a muslim. As an orthopraxic, rather than orthodoxic religion, to follow the Sharia is, in some senses, the very meaning of what it is to be a muslim. Islam focuses far more on correct religious practice than it does on correct religious belief or even on something like sin, which doesn't occupy anything like the position that it does in Christianity. He's correct that interpretations differ as to how various Muslims and Islamist parties believe that that ought to be applied in the political sphere, and in different countries "apply the Sharia" can be understood to have radically different meaning.

But that doesn't mean that we can't discuss what that law is. The vast bulk of the Sharia defines personal behavior and practice. Things like where ones hands ought to be placed during prayer. The conditions under which one needs to perform a minor or major ablution for touching oneself. The acceptability of alternative forms of ablution when you don't have water. Correct methods of food preparation and slaughter and so on. Most of these entirely non-controversial, and as a result when Muslims are asked if they want to follow the Sharia, such as in the polls that he cites, it would be a bit like asking a religious Jew if they want to follow the Kashrut.

The controversy is that there is a subset of Islamic religious law that needs to be enforced by a state rather than followed by individual Muslims. This includes the hudud, which perhaps the more infamous punishments of things like cutting off hands for stealing, stoning adulterers and executing apostates.

Even within this subset of those who actually want to apply these laws, however, there is a tradition in a number of Islamist movements that they want to do so in the context of fully enforcing other parts of the Sharia that in some sense would negate the need for the negative actions that necessitate the hudud. So, for instance, Islamist parties, going back at least to Abul Ala Maududi, have argued that in an Islamic state that enforces the Sharia, yes, you would cut off the hands of the thieves, but you would also collect and distribute the zakat, a form of charity that would theoretically negate the need to steal.

So while some Islamists have certainly said "no, just apply the hudud now" there is a popular trend in Islamist movements that instead suggests that in applying the Sharia as a whole you would create a society where seemingly harsh punishments for minor infractions start to make more sense. But even within that context most people simply understand the Sharia outside of a significant political context and simply understand it as their own religious law that governs the daily lives of muslims.