Preface: I am by no means an expert on this particular subject, and will only be speaking for the use of surnames in Sweden.

The surnames currently in use in Sweden have various origins.

Some (the most common ones, in fact) are patronymic - Karlsson, Eriksson, Johansson, etc., meaning "son of [blank]". This was for a long time the most common way to acquire a surname, and it is therefore the most common today, when the patronyms have become gender-neutral in usage and are inherited through the successive generations. Originally, daughters were given names such as Karlsdotter and Eriksdotter and so on - meaning "daughter of [blank]" - but while some of those names are still around today, they are far less common, since it's the son-form that has become accepted as the gender neutral.

(a quick aside - I believe the patronymic naming convention is still the norm in Iceland, with differing surnames depending on gender as outlined above. Someone feel free to correct me on this).

You also have the names of noble families. In the earliest days of Swedish nobility, the nobles had no surnames, but instead used sobriquets - Erik den läspe och halte (Erik the lisping and lame) and Ragnvald Knaphövde (Ragnvald Roundhead) were both ancient kings of Sweden (although the sources on at least Ragnvald are a bit scarce). However, as time wore on - and especially after the formalisation of the nobles as a separate societal class in 1279, and the first Swedish riksdag in Arboga in 1435 when the nobles were represented as a separate section of the riksdag - the naming conventions changed. At first they, too, used patronyms, but after being recognised formally as noble families and being issued with a heraldic shield, they began to acquire non-patronymic names - often derived from the symbols in their heraldry. And so you get noble families with names like Gedda, Banér and Uggla (Pike, Banner and Owl, respectively). For a while, you had them using both patronyms and their heraldic names at once - so you have cases like the king Gustav Vasa, who used the name Gustav Eriksson Vasa.

At the end of the 16th century, the noble class' usage of patronyms died out, and they almost exclusively used their heraldic names. You will also find that Swedish noble surnames use the "von" or "af" formation - "af" meaning "of", as in Henrik af Uggla, and so on. This is sometimes used to differentiate between branches of the same family, so you get someone whose surname is "Uggla af [blank]", with the blank being a geographical location, to set them apart from other members of the Uggla family.

(My personal favourite noble surname in Swedish is the family Natt och Dag (Night and Day) - they're still around today, and are the oldest contiguous noble family in Sweden)

Another surname-origin is that of soldier-names. With the introduction of the allotment system in the Swedish military, the need to differentiate between different soldiers in the same regiments, all bearing the same names (two Erik Karlsson's, for example) arose, and the soldiers were given specific sobriquets to set them apart. These names - personal name, patronym and sobriquet - were written down in the rolls to keep track of the soldiers. Inspiration for these soldier-names were taken from all sorts of things, and you get names like Blixt (lightning), Stål (steel), Modig (brave), Starke (strong) - but also Broms (a type of insect), Ljung (heather), Hjort (deer) and Fiskare (fisherman).

You also get names derived from geographical locations and features, as well as general nature-names. Berg (mountain), Ek (oak), Almgren (elm-branch), Björkman (birch-man), etc. These names were common in the merchant and middle class. This level of society also used trade-names, or a combination of trade and geographical feature names - I used to go to school with a boy named Smedberg (smith-mountain) for example.

Foreign influence shows up, even in the older Swedish surnames - we've borrowed from Germany, resulting in names like Möller (miller), and we've borrowed certain name-endings from France (the "-in" ending is purportedly French in origin) when that was in fashion, and so on.

Sidenote - Sweden has certain naming laws that dictate what is and what is not appropriate to have as names. These laws mostly regulate personal names (they're pretty basic - you can't name someone a brand-name, or a curseword, etc), but there are rules regarding surnames as well, mostly about which ones you are allowed to switch to. However, I don't know them well enough to speak about them here.

About soldier-names: Arne Ekman, Om soldatnamn, 1954

Swedish naming traditions in general

Also, university lecturer Sonja Entzenberg at Uppsala university has written a lot on Swedish surname usage, both in general and in particular about the patronymic system. You can find a list of her writings here

ETA: Editing in additional info I added below to the parent-post:

One thing I did forget to mention is the naming convention that is common in the Dalarna-region of Sweden, where you get surnames that seem to consist of personal names; for example Edvin Andersmats (Edvin, Anders and Mats are all male personal names, which are here pulled together into a personal name-surname combination) or Oskar Perols (Oskar, Per, Ola).

It is more clear in the name Oskar Perols, but the surname is formed in such a way as to imply belonging somewhere; just as the patronymic "-sson" names imply you are the son of a certain father, the "Perols" name implies that Oskar belongs to Per-Ola. "Ols" is just the northern dialect version of "Ola's". So in a way, it's a patronym as well - it just doesn't follow the son/daughter-formation common in the rest of Sweden.

Here with the Irish perspective:

Many Irish surnames today are prefaced by an “O” or “Mac” (or were at some point in history). These names are of older Gaelic origin, and may have numerous different origins. Firstly, the name could derive from an ancestor: An example would be today’s “O’Toole”. This comes from the Gaelic “O Tuathail”, which expresses descent from Tuathal, 10th century king of Leinster. Another origin could be from a place name: See today’s “Delaney”. Though not preceded by an “O” or “Mac”, it carried a prefix in older times as the Gaelic “O Dubhshlaine” meaning “of the black river Slaney”. This, like the previous example, exhibits a great degree of Anglicisation. Thirdly, the name could derive from an occupation. See today’s “O’Leary”, from the Gaelic “O Laoghaire” meaning “son of the cowherd”, or “McGuinness”, from the Gaelic “Mac Aonghusa”, meaning “son of the healer”. Finally, the name could have its roots in personal characteristics. Today’s “Kennedy” came from the Gaelic “O Cinneide” meaning “of the ugly head”. Sorry to burst your bubble, any Kennedys out there. I know that the above applies to Irish surnames, but I can’t speak for other cultures. I suspect many names are of similar derivation.

Sources:

http://www.dochara.com/the-irish/surnames/irish-surnames-of-gaelic-origin/ (Explains the composition of surnames)

http://www.rootsweb.ancestry.com/~irlkik/ihm/irenames.htm (Explains the origins of many modern Irish surnames)

This is a huge topic and I'm sure there's a great deal more that can be contributed on the topic, but to get you started, you can look at this list of previously-asked questions about surnames. In particular this thread looks like it covers quite a number of different cultures.