awk
The % is a modulus operator and NR is the current line number, so NR%2==0 is true only for even lines and will invoke the default rule for them ({ print $0 }). Thus to save only the even lines, redirect the output from awk to a new file:
awk 'NR%2==0' infile > outfile
sed
You can accomplish the same thing with sed. devnulls answer shows how to do it with GNU sed.
Below are alternatives for versions of sed that do not have the ~ operator:
keep odd lines
sed 'n; d' infile > outfile
keep even lines
sed '1d; n; d' infile > outfile