Odd Product!

A 2-digit number has an odd product if and only if both digits of the number are odd. Therefore, there are $5\times 5 = 25$ such 2-digit numbers.

The following figure shows the products of digits of all 2-digit numbers. It can be seen that there are $25$ odd products (highlighted yellow) when both the digits are odd.

If we need an odd product then the number should be of the form

XY --------- where X and Y both are odd .

We have 5 odd digits 1,3,5,7,9 Therefore the total possible numbers are $5^2$ = 25

This solution is for beginners not for experts. See @Chew-Seong Cheong solution for fast track answer.
1 can be combined with 1, 3, 5,7,9 to have desired result i.e 9 combination
3 can be combined with 3, 5,7,9 to have desired result i.e 7 combination
5 can be combined with 5,7,9 to have desired result i.e 5 combination
7 can be combined with 7,9 to have desired result i.e 3 combination
9 can be combined with 9 to have desired result i.e 1 combination
Total combination=9+7+5+3+1=25

More of a combinatorics problem, basically the only thing is we have to ignore all the even digits and then consider all the possible combinations with the odd digits which is 5*5 = 25 which is the answer...