Firstly , all these are kinds of enzymes that do particular kinds of things in living systems.
A kinase is an enzyme associated with energy production. It is involved in transfering phosphate groups from one molecule to another. Effectively it energises a molecule so it can go onto a new reaction.

Involved in energy and metabolism. Kin- kinetic, or to move energy
A ligase is a kind of enzyme that helps to join molecules togther. For example DNA is a string of nucleic acids. These need to be joined and a DNA ligase is used. The enzyme needs to be shaped to fit part of the DNA or RNA molecule.

A DNA ligase. involved in joining molecules together ; lig- ligate or join
Synthases are enzymes usually involved in the production of a particular kind of biochemical. For example Tryptophan synthase, helps in the the last step for creating Tryptophan. Or ATP-synthase produces a final ATP molecule.

ATP-synthase. produce a particular kind of biochemical molecule; synth- to create or make
How they all work together

note: DNA ligase can just use ATP with no kinases
Sources/references:
- http://en.wikipedia.org/wiki/Synthase
- http://en.wikipedia.org/wiki/DNA_ligase