What is Gibbs free energy? How is it useful?

The Gibbs free energy of a process is a measure of how readily that process will proceed on its own - a negative value means that the reaction is spontaneous, a positive value means that it is not, and a value of or very close to zero means that it's an equilibrium process.
 

It's equation is: G = H - TS
Where G is the Gibbs energy, H is the enthalpy change, T is the temperature that the process is at, and S is the entropy change.

Temperature here is in Kelvin, and a classic trick is to give you the temperature in celsius so you have to remember to account for it by adding 273.
Another easy pitfall is that since the entropy S is so much smaller than the enthalpy H, it will be in J K^-1 Mol^-1 - and H will be in kJ K^-1 Mol^-1​. So you need to put both values in the same units. Always remember to check that your units 'add up' correctly - this is just as important as your numbers.

Typically a question will ask you to find the enthalpy change, or entropy change of a reaction, then give you the value you don't have and ask you to comment on how likely the reaction is to occur through calculating this Gibbs energy. These questions are, when it comes down to it, just a bit of arithmetic.