Remotes are additional access points to your tractor's hydraulic system. Generally CUTs come without remotes (larger more expensive CUTs usually have at least one however). The main reason to have remotes is to run implements that require hydraulic power. Basic implements like a plow, box blade or bush hog do not require any remotes as they just operate from the 3PT hitch and (for a bush hog) the tractor PTO. However, if you wanted to get optimal use of a box blade, rake, rear blade etc you could install a top and tilt setup that would allow you to make adjustments to the top link length without stopping and getting off your tractor. Same is true of the ability to use a blade or box blade at an angle to crown a road. You can set that manually with the 3PT arms but using a hydraulic tilt function (run by a remote valve), you can do that from your seat.
You can install remotes singlely or in pairs. I have, for example, only one set of rear remotes on my tractor. That means I could run a hydraulic top link but not that plus the tilt function at the same time. If I installed another set I could do both. How many you need depends on what you intend to use them for but I think most folks would agree that two is a good number to have.
Remotes cost $$ and while the exact amount depends on your tractor and who installs it, it seems that $400-500 per remote is a minimum cost with some dealers getting a few hundred more per pair. (note: there is potential confusion regarding the numbers of remotes as each single remote actually has two lines, one for hydraulic fluid output and the other is a return. A pair of remotes therefore actually has 4 lines and connectors.) Each remote can operate one function though that can vary from something as simple as the toplink length to a complex tool such as a backhoe.
There are some hydraulic mavens on TBN who can clarify technical aspects of remotes further but the bottom line is that if you are going to do more than just hook up a mower to your 3PT hitch, then eventually you will want at least one and probably a pair of remotes.