"I'd like to know what's the difference between "come in on weekend "and "come in on a weekend"?" The first is very informal - I'd call it bad English - for "come in on A weekend" or "come in on THE weekend". It depends on the context of the preceding conversation which one is meant. Not only is it potentially confusing, but it doesn't sound very good. However, you will hear the plural, as in "Can you come in weekends? and that is standard and common. Often it's the question asked by an employer if they want someone to come in to work on Saturday/Sundays for a period. And yes, a weekend is a Saturday and Sunday. (not a Saturday OR Sunday)

Greg's answer is right. I'll add that if your employer asks "can you come in on the weekend?", it's not clear whether he/she means "Can you come in on BOTH days of the weekend?" or "Can you come in on ONE of the days of the weekend?" They probably mean both days, since as Greg says, "the weekend" technically means both days -- but people also often say "the weekend" when they're describing something that happens on only one weekend day. ("I forgot to buy bread when I was at the store last weekend" only implies that the person went to the store at SOME point during the weekend -- probably not on both Saturday AND Sunday.)