What is the difference between STD_LOGIC and BIT types?

BIT has 2 values: '0' and '1'.

STD_LOGIC is defined in the library std_logic_1164.This is a nine valued logic system.
It has 9 values: 'U', 'X', '0', '1', 'Z', 'W', 'L' ,'H' and '-'.
The meaning of each of these characters are:
U = uninitialized
X = unknown - a multisource line is driven '0' and '1' simultaneously (*)
0 = logic 0
1 = logic 1
Z = high impedance (tri state)
W = weak unknown
L = weak "0"
H = weak "1"
-  = dont care

Type std_logic is unresolved type because of 'U','Z' etc.It is illegal to have a multi-source signal in VHDL.So use 'bit' logic only when the signals in the design doesn't have multi sources.If you are unsure about this then declare the signals as std_logic or std_logic_vector,because then you will be able to get errors in the compilation stage itself.
But many of the operators such as shift operators cannot be used on 'std_logic_vector' type.So you may need to convert them to bit_vector before using shift operations.One example is given below:
--example of how to shift a std_logic signal : right shifting logically by 2 bits.
--count is std_logic_vector.
output <= To_StdLogicVector(to_bitvector(count) srl 2);
--to_bitvector converts Std_Logic_Vector to bit_vector.
--To_StdLogicVector converts bit_vector to Std_Logic_Vector.

Note :- Remember that use "BIT" logic type only when you are sure that the signals are NOT multi sourced.
 BIT_VECTOR is an array of BIT's.Similarly STD_LOGIC_VECTOR is an array of std_logic type.