The pgrep command, as others have stated, will return the PID (process ID) of processes based on name and other attributes. For example,
pgrep -d, -u <username> <string>
will give you the PIDs, delimited by a comma (,) of all processes whose name matches being run by user
<username>. You can use the -x switch before to return only exact matches.
If you want to get more information about these processes (as running the aux options from ps implies), you can use the -p option with ps, which matches based on PID. So, for example,
ps up $(pgrep -d, -u <username> <string>)
will give detailed information of all the PIDs matched by the pgrep command.