Binary is the number system computers use internally. It has two digits, 0 and 1, and each place is worth double the place to its right: 1, 2, 4, 8, 16, 32, and so on.
Why two digits? A circuit stores each digit as a voltage. If a wire only needs to be low or high, small amounts of electrical noise don't matter: anything near 0 V reads as 0 and anything near the supply voltage reads as 1. Transistors also behave naturally as on/off switches. Two reliable states means base 2.
Each binary digit is called a bit. To read a binary number, add the weights that sit over the 1s: 1011₂ = 8 + 2 + 1 = 11. To go the other way, see decimal to binary conversion.
Long binary strings are hard for people to read, so engineers usually write them in hexadecimal (4 bits per digit) or octal (3 bits per digit). The bits underneath are the same.
Binary isn't only for numbers. The same 0s and 1s are the false/true values of Boolean algebra, which is why digital logic can treat a bit as a number and as a logic signal at the same time.