If you have a low profile, you purposely don't attract a lot of attention. Movie stars often try to keep a low profile when they're on vacation, so they don't end up being followed around by photographers.
If you hate publicity or being in the spotlight, you probably keep a low profile. People with low profiles don't tend to speak in public, wear flashy clothes or crazy costumes, or get in loud arguments with their friends at restaurants. This term is also an adjective, as when you describe a low profile political campaign or a low profile writer who lives in your town. Low profile is fairly new, coined around 1970 in the US.