I know we can substitute a string using sed but how?

For example: sed 's/(old variable)/(new variable)/ details.dat

Am I suppose to put $old variable or whatever? Because I tried many times, it didnt work by putting $old variable. Am I suppose to enclose it with "" or ''?

Please help

Well, it matters. Is $old defined somewhere? Or are you trying to take an example script and mold it to your needs? If so, you may need to define it previously in the script, like:
old= < whatever you want it to be >

If that doesn't help, can you paste that part of the script here, so we can see how it's working?

your missing a closeing single quote.

should be
sed 's/(old variable)/(new variable)/' details.dat

One more thing you may do is put every variable in { }.
like :
sed 's/${Old_var}/${New_var}/g' FileName
Cheer Up,
Anshul