The French Defense is a very solid opening for black. It begins with 1. e4 e6 2. d4 d5.
French Defense: Advance Variation
Usually after d5 your opponent goes e5, then you go c5 for a pawn break. If dxc5 play Bxc5 and your position is better. If the usual move after c5 is played, c3, you play Nc6, stacking pressure on the pawn. Your opponent might play Nf3 defending the pawn, from here you can have an early pawn break with f6, this will be discussed later. You play a move that goes against opening principle but is fine in this case. You play Qb6 stacking even more pressure. Your opponent could play a risky move such as Bd3. Do not take the pawn. You can see why in the example below.
Rather than going for the pawn immediately play Nge7 or Nh6, the goal of these moves is to play Nf5 putting so much pressure on the pawn. If from Nh6 they play Bxh6 just play gxh6. From there you have attacks like Qxb2 (attacking the rook, forcing the knight to move), you have Rg8 controlling the semi-open file to maybe play Rg4 putting more pressure.
So we know the variations when we play Nh6, but what about Nge7? Well it blocks the dark-squared bishop opponent might just take on c4. Don't get greedy though, play Qc2 attacking the e5 pawn. After this you can eventually move the knight on e7 and the c5 pawn is stuck and free to capture.
In the example above you can see that black will win the pawn on c5. Now that was the advance variation, where white will push to e5 and all that good stuff happens.
French Defense: Advance Variation with early pawn break
In the early pawn break you play f6 after Nf3, if exf6 play Nxf6 your development is better than your opponent's.
French Defense: Exchange Variation
You can probably tell from the name what you do in this variation. This variation can occur on move 3, with exd5. You will then play exd5 and try to castle fast by developing your pieces.
In short play 1. e4 e6 2. d4 d5 to initiate the French Defense. If e5 play c5. If exd5 play exd5 and prepare to castle.
Puzzles
Try the French Defense: Advance Variation.
Try it with the Exchange Variation
Try it with the pawn break