Assault is the stage where ranged enemies fly around shooting at you and the goal is not to get hit.
Assault stages get EXTREMELY difficult the later the game goes, so if you are having trouble taking no damage on a later stage, replay the earlier levels and try to pop one in that area. For me, they always popped on the levels ending in "5". Level 5, 15, 25, 35, and so on.
Make sure you have a few smart bombs (full smart bombs if you're having trouble) in your inventory in case things get hairy. Smart bombs are great for these levels because not only will they destroy any enemies on the screen, they will also destroy enemy projectiles.
Here is a clip I recorded. As you can see, when I am about to take damage, I use the smart bomb, which destroys all enemies and bullets on the screen. If you have upgraded pets like I do here, they help a ton. The pets also took a bullet for me a few times. It does not matter if your pets are hit, only if you are.
There are not onscreen popups that says "PERFECT" like the Challenge and Dodge levels, and unfortunately do you not get a $1,000 bonus like the other levels when you complete them perfectly.