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Verilog Tutorial.
Digital Basics Tutorial.
Binary Multiplier discussion for 2 bits
Lets start the discussion with an example of two bit binary numbers. To explain the carry lets use both numbers as 3.

A = 2‘b11 (In verilog 2‘b stands for 2 bits wide binary number)
B = 2‘b11
The block diagram implementation of multiplier shows that the circuit requires four AND gates and two half-adders.

Refer half-adders circuit and verilog code from links below
Discussion of Multiplier circuit
Circuit Level Implementation of multiplier is shown below
A0B0
A0
B1
B0
A1
B1
B0
HA
HA
A0B1
A1B0
A1B1
Sum
Carry
Sum
Carry
4’b1001 = 9 in decimal  is equivalent to binary multiplication of 2’b11x2’b11.
A
1
1
Multiply B
1
1
Partial Product 1
C 1
C 1
1
1
Partial Product 2
1
1
Sum
1
0
0
1
Binary multiplier is very similar to decimal multiplication. The implementation is discussed in details with diagram and examples.