The AND gate functions the same way as a logical "and" operator. Thinking about 0 as false and 1 as true, the AND gate only outputs true if both inputs are true, otherwise it outputs false.
The OR gate functions as a logical "or" operator. It outputs true if one or more of the inputs are true, and false otherwise.
The NOT gate is known as the inverter because it outputs the invert of the input (ex. if the input is true, it outputs false).
The NAND gate is the AND gate followed by the NOT gate. As a result, the outputs are the inverses of what the outputs would have been for the AND gate. The output of a NAND gate is true if any of the inputs are false, and false otherwise.
Similar to an NAND gate, a NOR gate is an OR gate followed by a NOT gate. The outputs of a NOR gate are false if any of the inputs are true, and true otherwise.
The XOR gate (exclusive-OR) gives an output of true if either of the inputs are true, but not if both inputs are true. If both inputs are false, then the output is false.
The XNOR gate (exclusive-NOR) is the opposite of the XOR gate. If either, but not both, of the inputs are true, then the output is false, and true otherwise.