1
10100101
+11101101

Line up the columns.

2
10100101
+11101101

Start with the right most column (highlighted).

3
1 
10100101
+11101101
0

1+1=10 (since we are using binary, so one bit is carried over to the next column.)

4
1 
10100101
+11101101
0

Add the highlighted column.

5
 1 
10100101
+11101101
10

1+0+0=1, so there is no carry over. A carry over of zero is usually omitted.

6
 1 
10100101
+11101101
10

Add the highlighted column.

7
1 1 
10100101
+11101101
010

Carry over 1 to the next column.

8
1 1 
10100101
+11101101
010

Add the highlighted column.

9
11 1 
10100101
+11101101
0010

Carry over as necessary.

10
11 1 
10100101
+11101101
0010

Add the highlighted column.

11
11 1 
10100101
+11101101
10010

12
11 1 
10100101
+11101101
10010

Add the highlighted column

13
1 11 1 
10100101
+11101101
010010

Carry over the 1.

14
1 11 1 
10100101
+11101101
010010

Add the highlighted column.

15
11 11 1 
10100101
+11101101
0010010

Carry over as necessary.

16
11 11 1 
10100101
+11101101
0010010

Add the highlighted column.

17
111 11 1 
10100101
+11101101
10010010

Carry over the 1.

18
111 11 1 
10100101
+11101101
110010010

19
111 11 1 
10100101
+11101101
110010010

The final sum.