Exercises for arms?

The best way to acquire the ability to do pushups is to do pushups, if that's what you're after. Keep doing knee pushups until you get stronger, or start with wall/incline pushups (where your hands are on a surface higher than your feet) and work your way down to the floor. If you've got stairs in your house, use those; otherwise, walls, counters, tables, and chairs.

This is quoted directly out of one of our Gym fliers for "dorm room workouts"

Shoulder Press
Muscles used: Shoulders and arms
1) start standing with feet shoulder width apart and holding water bottles at shoulder height (picture shows 1 - 20 oz bottle in each hand)
2) Push the water bottles up by extending both arms towards the sky
3) lower back to staring position and repeat

Chair Dips:
Muscles used: Chest, arms, and upper back
1) start by placing your hands on each side of the chair, your legs should be bent in front of you and your bottom hanging off the chair
2) slowly lower your bottom towards the ground, go as far as your body will let you
3) using your arms, push back up to starting position and repeat

Bicep Curls:
Muscles used: upper arms and forearms
(again with the water bottles)
1) start standing with your feet shoulder width apart, holding the water bottles at your side
2) flex your elbows with palms facing up and bring the water bottles towards your chest
3) lower arms back to starting position and repeat

Plank (A true killer for me personally!)
Muscles used: entire body
1) start similar to push-up position only with your elbows on the floor instead of your hands
2) lift your body off the floor so that your ankles, knees, hips, and shoulders are in line, hold that position as long as possible.

If the 20 oz bottles get too light, you can always increase the weight by getting bigger bottles. I may be wrong, but I think a gallon of water is 8 pounds, so you can get that far without having to buy anything extra for exercise if you don't want to.