The Iron Gym is an inexpensive pull up bar that you can easily attach to a door frame and just as easily remove it. It uses leverage, so there are no screws and you don’t have to worry about damaging your door.
Check out this 2 minute video (click the play button once):
Chin ups (a variation of pull ups) are a great way to build biceps without weights. Use a reverse grip to emphasize biceps.
You can also do curls with a set of resistance bands.
Without equipment, you have to find something you can curl that weighs enough to challenge you.
Pushups (yep, pushups again) work your triceps as well as chest and shoulders. To target triceps more you can do “diamond” pushups. You basically do pushups with your hands close enough for your index fingers and your thumbs to touch or almost touch.
Tricep dips are another great bodyweight exercise for triceps. You can plant your feet on the ground or elevate them for an added challenge. Here's a quick video with feet elevated: