Greetings! This is a most interesting question before us, and it is an important one when considering the role of the German government and its various institutions in the lead-up to and during the First World War. Even more interesting is the fact that the Reichstag (or Imperial Diet), was one of two major constitutional bodies which represented (to various extents), the "parliamentarization" of the German Empire. It remains a topic of some debate whether or not either the Reichstag or its counterpart, the Bundesrat (or Federal Council), were actually successful in becoming parliamentary institutions of the German Empire. Note that this response is by no means an in-depth look at the history of the Reichstag from 1871-1918, but it should help to paint a general picture of how this government body acted and evolved over those years. Let's begin.

Note

And, what power did it have over the different states within Germany?

To address this question first, it is necessary to realise that the laws enacted by the Imperial government in Berlin superseded the legislations of any of the individual member states of the German Empire. Such a provision is dealt with in Article 2 of the 1871 Constitution of the German Empire:

Within said territory, the Empire shall have the right of legislation according to the provision of this Constitution, and the laws of the Empire shall take precedence of those of the individual states.

So whilst the Reichstag had no direct power per se to overrule the laws of the member states (for example, Bavaria with its minority separatist movement), it did by way of the consitution have the ability to vote on laws which would automatically replace or take precedence over the pre-existing laws of the individual states.

Technicalities

Firstly, let us consult the Constitution of the German Empire for the constitutional rights and responsibilities of the Reichstag in tandem with the Federal Council. Under Section V, dedicated to outlining the composition and duties of the Reichstag, we find the following:

Article 23

The Imperial Diet shall have the right to propose laws within the jurisdiction of the Empire, and to refer petitions, addressed to it, to the Federal Council or the Chancellor of the Empire.

And... that's pretty much all we have in terms of the constitutional rights and duties of the Reichstag. The Imperial Diet could not dissolve governments (but it could itself be dissolved by the Bundesrat, with the consent of the Deutscher Kaiser), nor could it elect the Reichskanzler (Chancellor, who was also appointed by the Kaiser and presided over the Bundesrat). For that matter, the Reichskanzler was not even responsible to the Reichstag, and had no obligation to abide by the opinion of it as a governing body. However, the Reichstag did have a say in the passing of legislature, as detailed in Article 5:

Article 5

The legislative power of the Empire shall be exercised by the Federal Council (Bundesrat) and the Imperial Diet (Reichstag). A majority of the votes of both bodies shall be necessary and sufficient for the passage of a law.

The Reichstag was also responsible for, alongside the Bundesrat, overseeing the annual national budget and expenditure, something which caused great concern to Kaiser Wilhelm II and his naval admirals during the arms race of 1906-1912. What the Reichstag was in essence was a national forum with elected representatives of the people, and that enabled political parties in Germany to grow in power and (more often than not), obstruct the wishes of the Imperial government or even the Kaiser themselves. As I go into further in this thread, the Kaiser was not that powerful as well constitutionally, but did have a fair bit of influence in reality.

So how did this ambiguously defined Reichstag hold up in the leadup to and the beginning of the First World War?

Reality

From the creation of the German Empire and its constitution, the Reichstag slowly but surely became a more influential, but not necessarily powerful, part of the government. As the representation of the landed gentry and nobility began to decline in the Reichstag, replaced with the various political parties representing the various classes of contemporary Germany, the opinion of the Reichstag became a massive factor in the considerations of the Bundesrat and also the Kaiser. Rather interestingly, the Reichstag did take part in the Zabern (or Saverne) Affair of 1912, when inflammatory insults and military escalations from a Second Lieutenant in the city of Saverne (in the province of Alsace-Lorraine, then a German possession) caused a major uproar in the Reichstag. For the first time in its history, a vote of no confidence was called, and Chancellor Theobald von Bethmann Hollweg was...unaffected? Indeed, one would think that a vote of no confidence would surely lead to some sort of repercussions on a government official, but under the constitution and the standing orders of the Reichstag, there was simply nothing more than could result from such a move. The Reichstag representatives at the time did not take any further action, and Bethmann-Holweg was allowed to resume in his capacity as Reichskanzler.

What the Reichstag did possess however, was a growing influence on the German public, which concerned the conservative parties and the Bundesrat to no end. With the politicisation of the German populace, the proceedings of the Reichstag were very much of importance to the average German citizen, regardless of which state or social class they belonged to. It was for such reasons that the Imperial government had to tread carefully during the Naval Arms Race of 1906-1912, when new funding for dreadnoughts and other vessels for the Kaiserliche Marine (Imperial German Navy) had to be presented to the Reichstag as expenditure on national defence rather than expansionist capabilities. Recall how the Reichstag had the ability to check and disapprove of the federal budget; this was the particular power which the Kaiser and his officials absolutely loathed the assembly for. Even before the 20th century, the growing Socialist parties in the Reichstag worried the conservative ruling elite. The largest of these parties the Social Democratic Party, (SPD) would come to dominate the assembly by 1912, being the single largest political party with control of over a third of the 397 seats in the Imperial Diet.

Another critical moment of the Reichstag occurred in 1914, when Wilhelm II and his war ministers feared that the Reichstag would heavily disapprove of the decision to support Austria-Hungary, and for that matter to get involved in a war that did not seem to concern them. In order to convince them, the Kaiser and his advisers continually pushed the Austro-Hungarians to portray the Serbians as the perpetrators of the archduke's assassination, and that their declaration of war would be in retaliation for such a heinous crime. When the time came, the Reichstag was compliant in accepting the German government's call for a general mobilisation, and went ahead with its support for the war.