A bike chain tool “breaks” a chain by screwing a pin into the rivet joining two links, forcing it out and thus separating the links. To connect two links, you use the tool to drive a new rivet, known as a connector pin, into the empty rivet holes.

The best bike chain tool: Birzman Damselfly Universal (multi purpose, stylish, wide wing-shaped crank is good to hold and feels sturdy, compared to other slender options)
The second best bike chain tool: The Park Tool CT-3.2 Chain Tool and Topeak’s All Speeds Chain Tool (with narrow, ergomanically deficient handles with poor grip)
The best upgrade for bike chain tool: Pedro's Pro Chain Tool (comes with a patented “retracting pin guide” technology, expensive)

Apart from these options, there are also dual-purpose chain tools, which you can keep in a pocket and use on the trail as well as at home. 
For example, you have Park Tool’s CT-5 Mini Chain Brute Chain Tool, Pedro’s Six-Pack Chain Tool, and the Lezyne Chain Drive.