I recently joined this manga and anime forum and stumbled across this term numerous times. People reviewing different parameters of animes and calling that a "typical" slice of life anime.
Is this pretty common sort of anime?
What does this mean?
This term actually originates outside of anime, but it's rather common for anime to have slice of life elements, so it's a common term in the anime fandom. Toshinou-san's answer covers the general sense of the term very well, so in this answer, I'm going to try and fit it more into a critical framework, which can be helpful for understanding whether a given individual work is slice of life. Like everything in literary and film criticism, this is all pretty subjective, so people will differ on whether they think an individual series qualifies as slice of life under this definition or not, but the concepts should be pretty universal.
You can think of all media as falling on a spectrum between theatrical and naturalistic. Theatrical works use dramatic conflicts, larger-than-life characters, architected plotlines, and other artificial manipulations of events and causality to tell a story which the audience will find interesting. By contrast, naturalistic works try to tell stories which develop the way things tend to develop in real life. They can still have conflicts, plots, and interesting characters, but those things will all tend to be truer to life and less artificed.
Here are some examples. (It's hard to find examples of naturalistic anime which aren't also slice of life, because, well, it's anime; it's inherently unnatural. So I've used American films as examples of naturalism, because I want to be clear that slice of life is a subset of naturalist, not an equivalent term.)
Note that this has nothing to do with genre. Some fantasy and science fiction works are naturalistic. Some works which never reach outside the bounds of real life are nevertheless theatrical, as we see with Hana Yori Dango and Rurouni Kenshin (to an extent).
So what does this have to do with slice of life? Wikipedia defines slice of life as "the use of mundane realism depicting everyday experiences in art and entertainment." Essentially, slice of life works are naturalist works which are about mundane, everyday experiences. They might be naturalist works about everyday experiences somewhere where every day is very different from what audiences know. That's still slice of life; the important thing is that the experience is mundane to the characters, within the setting. For this reason, The Hurt Locker is not slice of life. It is naturalist, and in some sense it's about everyday experiences in a war zone, but the war zone setting is not mundane to the characters at all. One could argue that The Best Exotic Marigold Hotel is slice of life, though I consider it more of a drama. (Pure slice of life is vanishingly rare; almost all slice of life shows have drama or comedy elements, but they're the kind of little dramas and jokes that arise in mundane life, not huge scripted events.)
The classical examples for slice of life in anime are shows like K-On, Yotsuba&!, and Ichigo Marshmallow. These are slow-paced shows about everyday life. We see the characters go to school, go shopping, go on trips, spend time together at home or in a club after school. There are jokes, but they're either jokes the characters themselves make, or they're jokes arising from everyday situations. Slice of life shows don't usually use high concept jokes like having a Pixxchu pop up and get beaten to death, or having former Japanese prime minister Junichiro Koizumi suddenly appear above Tokyo on a flying saucer, or telling us halfway through the show that the heroine's desire to be an idol stems from her past life in a vanished human civilization which was exterminated by aliens in the mists of prehistory. Slice of life shows also can have drama, but it's quiet, everyday sort of drama; K-On, for example, has some drama around Azusa's feelings when her upperclassmen graduate. Aria (discussed below) has drama around Alice's feeling of separation when she becomes a Prima and doesn't have time to spend with her friends. A slice of life show wouldn't usually have the heroine's ex-boyfriend suddenly back in town and looking to rekindle his feelings just as the heroine is about to move her relationship with her new boyfriend to the next level; that sequence of events is just too convenient for the writers.
Aria, Yokohama Kaidashi Kikou, Haibane Renmei, and Kiki's Delivery Service are usually also considered slice of life, even though the first two are science fiction and the last two are fantasy. Much of what we see in these shows is mundane and everyday to the characters. There are no dramatic conflicts, no epic clashes, nothing extraordinary or unusual (from the characters' standpoint); its just people (and Martian gondoliers, and androids, and grey-winged angels, and witches) going about their everyday lives. Haibane Renmei and Kiki's Delivery Service do actually have plots, and some of the events in Haibane Renmei are unusual from the characters' perspectives, but all the events of the plot come about in the course of a natural, everyday life in the world of the series.
In anime, the concept of iyashikei is also related to slice of life. Iyashikei is, as far as I know, completely unique to anime and rather characteristic of Japanese culture in general. Many slice of life anime have greater or lesser elements of iyashikei; Aria and Yokohama Kaidashi Kikou are practically the epitome of the genre. Not all slice of life is iyashikei, and not all iyashikei is slice of life; however, the two are closely bound, and the desire for iyashikei may have driven the prevalence of slice of life.
From Wikipedia:
Slice of life is a phrase describing the use of mundane realism depicting everyday experiences in art and entertainment.
Generally in anime/manga, slice of life shows are shows about everyday life. Usually this rotates around school life in anime - like Lucky Star, Nichijou or K-On! but this can extend to other stories such as people living in apartments, working a day job, etc.
Slice of Life shows are usually plotless as they are mostly day-to-day happenings (usually with humourous effects).