Can you leave a past employer off your resume?

Answer

Unless this old job was at the prison laundry, or your potential employer is the CIA, I wouldn't worry too much about background checks.
You're under no obligation to list all your previous employers on your resume. In fact, it's generally a good idea to only list jobs that you think help make you more qualified for the position you're applying for.
The potential risk in leaving some of your job history off your resume is that the employer will notice the "hole" in the timeline and it will look suspicious. They might wonder what you were doing during this period. But in your case, since you only had the job a month, this risk is minimal.
P.S. Check out this page at privacy.org for more serious, detailed information about background checks.)  

Answer

Be careful about leaving anything out for the most recent ten year period. For the past two assignments give year and month of beginning and end of assignment. Beyond 10 years or so, you can be very broad. But, don't quibble or equivocate or downright lie. It may be considered grounds for termination. For higher level assignments, always assume that a full check of all credentials and references will be done. Otherwise the recruiting firm really hasn't serviced the client properly. Any negatives (and most of us have at least one glitch in our careers) can be handled verbally (best) or in the cover letter.  

Answer

Be careful. It makes a very bad impression indeed if it emerges that you are trying to hide something. Don't make the mistake of assuming that a potential employer is stupid and won't spot a gap in your record.

It depends on what type of employer it was .For example let's say you worked at a fast-food chain when you was 18 and worked there for 2 and half years. You are now age 30 and just graduated from college with a masters in Biology. You are trying to find a position that will relate to your masters degree. Do you think that a prospective company cares what you did 22 years ago? No. Your position at the fast-food chain has no relevance to that position you are trying to seek