Death rays, clockwork guns, and steam-powered cannons are some examples of the weaponry that can exist in a steampunk tale. What weapons you use depends on whether you decide to make things realistic or fantastic. My general rule is to not make anything up when something that already exists will serve the purpose. Some basic research will show the vast catalog of weapons that people used in the 19th Century, in details more than enough to satisfy the creative mind. Changing the names is OK, by the way.
Take the repeating rifle for example: a standard rifle designed in the late 1800s to improve on the breech- and muzzle-loading rifles of the time. There are four types of repeating mechanisms at use in this firearm -- bolt action, lever action, revolving, and pump action. Lever-action repeating rifles are the one seen in a lot of Westerns, where the shooter fires, pulls the trigger assembly down, and fires the new bullet that's been levered into place. This gun serves the purpose of giving soldiers a long-distance weapon that fires multiple bullets before reloading. And is a real weapon that can fit easily into any steampunk work.
The former Royal Small Arms Factory of Enfield, London crafted numerous ubiquitous weapons that fall under the auspices of traditional steampunk with its Victorian setting. The Enfield revolver and the Lee-Enfield rifle, two examples that bear mentioning, were in service well into the 20th Century for the British military. This is one of those cases, however, where you can't use the actual names of the guns unless you're writing a traditional steampunk or historical fiction.
Going forward with more esoteric weaponry, such as those with clockwork firing mechanisms, it's always worthwhile to consider how the design might potentially function. No one's really going to build a gun with a clockwork firing mechanism because traditional mechanical versions do the job they're supposed to. There's no point to doing anything with the level of complexity a clockwork mechanism requires. However, considering how the machine might work can add a certain veracity to your writing that you might not get simply from saying "a clockwork gun." It also helps if you have a MacGuyver-ish character that has to repair it in the context of the story.
Steam-powered cannons and other heavy guns are easy answers for determining how they fire. Steam generated by the attached boiler is compressed until the pressure increases to a suitably high level behind the cannonball/shell/whatnot and, when the pressure is released, the shot explodes from the cannon.
Then, there's the stuff of pure fantasy -- death rays, dream extractors, and whatever else you want to make up as weaponry in your story. The sky's the limit really. Just remember to think about how it operates. Details are good.