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Whenever I play Chess, a mix of numbers and characters appear.

There are the normal a-h movements, but some moves are nf3, bb5, qxb5.

What are those?

http://www.chess.com/chessopedia/view/algebraic-notation

The a-h are not "movements" and neither are the 1 to 8 up the board.

a thru h are what we call "files". Each file is a vertical line of squares on the board. a is always on White's left, so if you are Black, it will show as h downto a.

1 thru 8 are what we call "ranks". Each rank is a horizontal line of squares on the board. 1 is always where White's pieces start (2 being White's pawns), so if you are Black, you pawns will advance forward from 7 downto 1.

Think of the squares on the board like graph paper in algebra class. The x-axis was the first value in a point, the y-axis was the second. So the point (3, 2), you had to go to 3 on the x-axis, and then up 2, and you had your point.

It's similar here, only you use a letter and a number to display your coordinate. So the dark square 2 squares in front of where White's Queen's Pawn starts on is "d4" (The square where the "d-file" and "4th rank" intersect.

When you move a pawn forward, and not capture anything, you simply put the square it went to. So a common first move for White is 1.d4

That's always the last 2 characters of any move other than Castling. Anything prior to that is piece indicators, clarifications, and capturing.

Pieces: The pawns are just called that. Pawns. "Pieces" refer to everything else. K = King, Q = Queen, R = Rook, B = Bishop, N = Knight. Pieces are ALWAYS ALWAYS ALWAYS indicated by a CAPITAL LETTER. Therefore, if White moves his King's Knight on the first move to f3, it is shown as 1.Nf3, NOT EVER 1.nf3. The reason for this is simple. Let's say White has a Bishop on d2 and a pawn on b2, and Black just took the Knight on c3. How do you distinguish "b-pawn takes c3" and "Bishop takes c3". Pawns are always lowercase, pieces are always capital. So bxc3 is b-pawn takes c3. Bxc3 is Bishop takes c3.

Capturing: Based on the last sentence, I'm sure you figured out that x means captures. So 24.Bxf6 means for White's 24th move, he did Bishop takes f6 (Whatever Black piece or pawn was on f6, doesn't matter), whereas 24.Bf6 means White moved his Bishop to the empty square, f6

Clarifying: While it can occasionally happen with Queens, and even rarer Bishops, very common is having the problem with Knights and Rooks of "Which one went to the square?". Let's say you have a White Rook on a1, and a White Rook on f1, and NOTHING IN BETWEEN. To say "17.Rd1" is ambiguous. Which rook went to d1? The Rook on a1, or the Rook on f1? You indicate that with an extra letter between the "R" for rook and the "d1" square. So "Rad1" is you moved the Rook on a to d1. "Rfd1" is you moved the Rook on f to d1. "Raxd1" would mean you used the Rook on a to capture something on d1, and "Rfxd1" would mean you used the Rook on f to capture something on d1. Let's say you have 2 rooks or 2 knights on the same file, like Rooks on e1 and e5, and one of them goes to e3, or Knights on e1 and e5, and one of them goes to d3. Then you use the rank number, like R5e3 or N1d3.

Black vs White: If you are listing a single move in a game (like when you read a chess book), 3 periods (or dots) means Black moved. So 17 a4 or 17.a4 means White moved a pawn to a4. 17...a4 means Black moved a pawn to a4. This is critical. Let's say White's pawn was on a2, Black's on a5. Either player could put their pawn on a4 on their next move, so clarification is critical.

So if you are to notate the first 7 moves of a game that starts with the poisoned pawn French, it would look like this:

1.e4 e6 2.d4 d5 3.Nc3 Bb4 4.e5 c5 5.a3 Bxc3 6.bxc3 Ne7 7.Qg4 Qc7

(Notice the difference between "Bishop takes c3" by Black on move 5 versus "b-Pawn takes c3" by White on move 6.