grep -c -v '^$' filename
If you want to, as well, exclude lines which contain only spaces:
grep -c -v '^ *$' filename
Notes (from the grep man page):
-c, --count
Suppress normal output; instead print a count of matching lines for each input file. With the -v, --invert-match option (see below), count non-matching lines. (-c is specified by POSIX.)
-v, --invert-match
Invert the sense of matching, to select non-matching lines. (-v is specified by POSIX.)
Anchoring
The caret ^ and the dollar sign $ are meta-characters that respectively match the empty string at the beginning and end of a line.
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