Laymans guess to determine how many hashes must be computed to you find a specific value I have a question about bitcoin, lets say that the output is uniformly random and that it is 256 bits in length.  The output of the hash function y=H(t) begins with 25 zeros - how many times would one have to call the hash function to get something that starts with 25 zeros?

Secondly, how does one verify the solution and how many hashes would be attempted to solve it?
It's pretty simple. Since the hashes are uniformly distributed, 25 zeroes is as likely as any other possible value for the first 25 hexadecimal digits. There are 16 possible values for each hexadecimal digit, so the number of possible combinations is 16^25, or about a 1,2676506×1030. This also means that on average, one in 1,2676506×1030 hashes will start with 25 hexadecimal zero digits.

To verify the solution, you just compute the hash once and check if it starts with the required number of zeroes. So, it's 1030 times easier to check the hash than to find it.