If your basement floods or becomes damp during wet periods, first try directing rainwater away from the house by changing your gutter downspouts. If that does not solve the problem, a sump pump may be the solution.
There are two basic kinds of sump pump installations. If you have no drainpipes under the basement designed specifically for handling rainwater, then dig a hole at a low point of the basement and install a perforated pit liner (you may have to perforate it yourself using a 3/8-inch spade bit). Water under the basement will slowly percolate into the liner.
If your house has rainwater drainpipes, it probably also has a pit already dug for a sump pump, with a drainpipe running into it. Purchase a nonperforated pit liner and cut a hole for the drainpipe.
A wise add-on is a rechargeable battery backup unit, which will power the pump if a heavy storm causes a power outage. A pump-failure alarm is also a good idea.
About half a day
Sledgehammer or electric jackhammer, cold chisel, drill, masonry bit, spade bit, hole saw, PVC saw, torpedo level, screwdriver, groove-joint pliers
Basic carpentry skills, cutting and joining PVC drainpipe
Locate a low spot where the pump won't take up usable space; if necessary, install a GFCI receptacle
Sump pump, pit liner, PVC pipe, primer, cement, check valve
Use the pit liner to mark the right-size hole to cut in the basement floor. Using a masonry bit, drill a series of holes along the perimeter. (A concrete basement floor is usually 3 inches thick.) Chip out the concrete with a sledgehammer and cold chisel or an electric jackhammer.