Here are some of the comics I like where you can jump right in and quickly figure out the story (if there is one):
Questionable Content. My favourite online soap opera. Exploring the lives of a group of twenty-somethings in a world with some resemblance to our own. Updated Mondays to Fridays. It's better to have read the whole thing from the beginning, but you can quickly catch up even if you don't.
Kevin and Kell. The adventures of anthropomorphic animals in a world that mirrors our own. It is better to have started from the beginning, but the "About" and "Cast" pages will fill you in on some of the details and help you catch up if you don't feel like reading the whole thing. Updated daily with an extra large strip on Sundays.
Toothpaste for Dinner. Random humour. Updated daily.
Saturday Morning Breakfast Cereal. Random humour, sometimes not funny, sometimes very much so. Updated daily.
Nickyitis. Often quite funny. Updates weekdays. Three days a week it's a comic about the titular Nicky, and two days it's single panels.
Cat versus Human. Cat humour. updates irregularly, but usually at least twice a week.Posted in blog form.
Wasted Talent. Updates once a week. Glimpses of the artist's life (she's an engineer and drawing is a hobby). You can jump in anywhere, but it's better to have read from the beginning.
xkcd. Billed as "A webcomic of romance, sarcasm, math, and language". Sometimes you need to be a science geek to understand it, but more often you just have to understand sarcasm. Updates Mondays, Wednesdays and Fridays.
Piled Higher and Deeper. Academic humour. Updates irregularly, but worth waiting for, especially if you're a grad student slaving away in teaching assistant and grant refusal hell.