This is a tricky question and really depends on the community in which you ask it.
Frankly, there are no real qualifiers. Some people think that unless you play only on hardest difficulty, you're not a "real gamer". I've seen some act like if you aren't a competitive player, you're not a "real gamer". Personally, I think anyone who loves to play video games is a gamer.
On a side note, I find it rather humorous when young players (20 somethings and younger) try to trash talk me because they beat me or because I'm not as good as them. Mostly because I was playing games a decade or so before they were ever born (I started in 1988) and I've just gotten to the age where I don't care about competition or trying to prove I'm the best. With my family responsibilities, I have limited time during the evenings to play and prefer to focus on accomplishing game objectives and having fun.