What is the name of the camera feature that can take consecutive shots quickly?

continuous shooting.

Cameras which can do that are DSLRs and mirrorless interchangeable lens cameras. Your budget isn't enough, unless you can find a used camera, go have a look at keh.com

Some entry level DSLRs can manage about 4 to 5 frames per second, some mirrorless systems as much as 11 frames per second, more advanced DSLRs (professional cameras which cost thousands) can manage 10 or more per second.

Night vision, no way. Sorry.

Waterproof, perhaps the Nikon 1 AW1, but your budget is way out.

- *Burst/continuous shooting* is for shooting in quick succession.

- *Bracketing* allows you to shoot the same scene with slightly different settings, to either pick the better option, or to stack them together for HDR or other effects.

- A memory card with a *faster write speed* allows you to save files more quickly (or at least as fast as the camera's image processor allows). More expensive cameras generally have faster processors.

- *Shutter lag* is the delay between when you fully depress the shutter and when the camera actually takes a photo. Most cameras already have low shutter lag, except maybe cheapo compacts. There may be extra delay needed to acquire focus and exposure, which is what actually slows most cameras down.

Everything you want except for night vision. That feature was more common in old camcorders. Today, you have to customize a camera to do that.

http://www.bhphotovideo.com/c/search?Ntt=ricoh+wg&...

Burst shooting

Continuous shooting

Interval shooting