Kinds of cases:
IP litigators include those who not only litigate patent but also copyright cases in federal court, or trademark and trade secret cases at either the state or federal level (for diversity), or even litigate contracts such as IP licensing agreements or NDA's at either the state or federal level. IP litigation can also extend to appellate work, depending on the firm and the clients.
IP litigators could also litigate infringement or other unfair trade practice in a 337 case (unfair trade in imports) at the International Trade Commission before an ALJ. (Interesting three-sided litigation: domestic complainant, foreign respondent(s), and then the ITC's own attorney representing the US government's interest.)
Then there's litigation before the Trademark Trial and Appeal Board or the Board of Patent Appeals and Interferences: During prosecution of a trademark application, someone may initiate an opposition proceeding to oppose registration of the mark. Or a granted registration may be challenged by someone through a cancellation proceeding. Oppositions and cancellations are litigation, and follow the same basic format as a court case including discovery, presentation of evidence, and argument. The difference is these are before the Trademark Trial and Appeal Board instead of a court and most of the litigation can be handled without leaving your office -- both conducting discovery and presenting evidence via paper (or, these days, electronically I presume) and over the phone. Challenges to patent prosecutions come in the form of Interference proceedings, which I presume also follow the same litigation style format as do their Trademark counterparts. You must be registered with the respective agency/board to either prosecute trademarks or patents or litigate them before the boards, which for patents means you must have passed the patent bar.
Chances are, you wouldn't handle such a proceeding unless you were already doing the prosecution work, but that could be the case. Particularly for IP boutiques, prosecution is the "bread and butter" of the practice that provides a steady income. When I was with a boutique, there were a number of patent attorneys who didn't litigate, but I'm pretty sure there were no patent-qualified attorneys doing litigation who didn't also do prosecution (with the possible exception of the name partners).
Patent vs. Non-patent
It sounds like you are patent qualified or expect to be -- since you mention contrasting prosecution with litigation--but, with respect to patent vs. other IP, I'll mention that while many firms do keep their patent people separate from other IP, there are firms where that's not the case. The skills required to draft a good patent, while considerable, don't necessarily pair up with the skills necessary to marshal the facts of a case into a compelling brief. Also, to do a good job litigating, you need to be able to translate both the law and the art into language the judge (or possibly jury) can understand. So it can be helpful to have a an attorney who isn't patent-barred on the team, but that person needs to be a quick study and able to learn both about the art and the patent law at issue from the patent attorneys on the case.
Kinds of activities within litigation:
This is also going to depend on your level of expertise and experience, as well as the firm you're with.
When you're new to practice, you may initially find your finely honed trial skills, including your mastery of the rule in Queen Caroline's case, will be mothballed while you steep yourself in discovery grunt work -- important but grunt nonetheless: filling out interrogatories and responding to requests for production and drafting the same for the other side, occasionally interrupted by motions to compel and subpoenae. With more experience, you may be graduated to making charts identifying and succinctly but innocuously describing documents you've been asked to produce which are attorney-client privileged or work product. A more interesting side of "grunt" preparatory work, however, can be going on-site to the client's manufacturing plant to get a first-hand understanding of a process at issue or to help pull out responsive documents for a production request. This can involve all day or overnight travel across county, or a few hours to a place in your home town.
As you gain experience you'll likely be drafting memoranda and more varied motions and briefs. And, depending on the firm, you may be sent to argue a motion or brought into the courtroom to assist at trial either sooner or later. Ideally, eventually you'll be actually examining witnesses and making arguments to the judge or jury, and even taking first chair, but how soon that comes about will vary by firm. (Some firms tend to hold back on bringing along new attorneys in courtroom work. Written work can be reviewed and redone if necessary, but not presentation of evidence or argument at trial.)