Chess is one of those games that is easy to learn, but hard to master. Think of checkers, but on steroids.
In chess, there are 16 pieces that you control on a 8x8 square grid, or a 64 square grid. The grid consists on the classic checkerboard pattern. So if you have a checkerboard, it doubles as a chess board. There are a total of 32 pieces that can be controlled, 16 to each player. If you cannot tell, chess is for two players at a time. There are six different types of pieces. Each player will have eight pawns (the basic cones), two rooks (or towers), two knights (or the horses), two bishops (the more fancy cones), one queen (usually the one of the two that does not have a cross on it some where), and one king (the one with the cross). The pieces will be set up in the two rows closest to the player (to keep this simple). All of the pawns will be in the row farther from you, and the special pieces will be in this order from left to right: Rook, knight, bishop, queen, king, bishop, knight, rook if you have the white pieces, the black pieces are the same, except the queen and king swap.
Now that the board is set up, what is the objective? The objective is to "capture" the opponent's king via Checkmate. To capture a piece you put your piece where the other piece was and remove the one piece that was in that spot. To achieve checkmate, one must make it to where the king is in position to be taken without a way to escape or another piece to protect or destroy the attacking piece. This sounds easy, right? Not really... Although there is many way's to achieve checkmate, the opponent is trying for the same goal.
So now to how the pieces move. Starting with the pawn, the pawn can move two spaces if in the starting position, afterwards, the piece can only move one space at a time, only forwards. The pawn is also special because the pawn can only attack diagonally forwards, along with this, if the pawn reaches the opposite side of the board (the squares closest to the enemies), the pawn can become any of the special pieces other than the king. The rook can move any amount of spaces forwards, backwards, and side to side, not diagonally. Bishops can move in all diagonals, and not straight. Queens can do what both the rooks and bishops can do in one piece. Knights move in an L shape, one in a direction, two in another, and is the only piece that can jump other pieces. The king can only move one space in any direction.
Finally there is a few more things mechanically, that needs to be stated. First, the white (or lighter colored pieces) will always go first. Second, kings can take pieces only if taking that piece doesn't result in being in a direct path for capture. Basically, if a there is an opponent piece that can move to the space where that take able opponent piece is at, the king cannot take it. Thirdly, you cannot move your own pieces out of the way without wasting a turn (basically if you need one of your pieces in a spot where another one of your pieces reside, move that piece first, this also includes the inability to take one's own pieces). Lastly, pawns can move one space forward at the beginning, you do not have to automatically move forwards the pawn two spaces.
So now you are ready to play. A few tips for the road. First, always look at your opponent's plan. If you tunnel-vision on what you want, you can have unnecessary casualties.Second, never corner your king. Basically leave an area of escape for your team. Pressure is great in this game, the more aggressive you play (without being reckless), the more mistakes you can force out of your opponent.
I know that this article is far from perfect in describing how to play, but there are multiple places you can go to learn more. YouTube is a great resource, along with chess.com, an International chess gaming site in which there are lessons you can take, professionals you can watch, and bots to hone your skill. You can even play real players in real time here as well. Watching very skilled players is also a good thing to do as well.
Enjoy your newly found past time!