As stated, that's a microwave antenna used for point to point communications.
The microwave systems provide a high bandwidth connection. Depending on what's residing on the tower, it could mean anything:
If it's a cellular tower, they use the microwave links to tie the cell site back to the mobile switching center.
If it's a broadcast tower, it's used as a studio <-> transmitter link. The link could be carrying broadcast audio, TV signal as well as system control. They often operate in both directions, hauling engineering data streams back to the studio.
If it's the phone company, it's providing services between central offices.
If it's a LMR/Public safety tower, then it can be used for carrying transmit and receive audio between dispatch centers and the radios sites.
As for the antenna, the back (right) half is the parabolic reflector like what you would normally think of on a satellite or microwave dish. The other half (left side) is the protective cover. Under that cover is the feed horn that feeds the signal to/from the focal point of the parabolic dish.
Since these feed horns can be fragile and can be affected by weather (think ice), the are often protected with these covers. The covers are an RF transparent material, like plastic, fiberglass, fabric, etc.
There are online tools where you can enter the location of the tower and it will show you what links are there, where they go and who the belong to, but only if they are FCC licensed. Not all microwave links require licensing with the FCC, some frequencies are "license by rule".