NOT gate -
This gate inverts the input and places the inverted value at the output, If you apply a 1 at the input the output will be 0 which is the reverse of 1, and vice versa.
NOT Gate Tutorial
AND Gate -
This gate ANDs the two inputs A and B, The Output is high only if A AND B both are high. There may be an AND gate with many inputs, only if all inputs are high the output will be high.
AND Gate Tutorial
OR gate -
This gate ORs the two inputs A and B, The Output is high if either A OR B is high. There may be a OR gate with many inputs, if one or more inputs are high then the output will be high.
OR Gate Tutorial
XOR gate -
This gate XORs the two inputs A and B, The Output is high only if either A or B is high, output goes low if both inputs are high or low. There may be a XOR gate with many inputs that may work as XOR or Parity Gates.
XOR Gate Tutorial