Ageplay is a type of roleplay in involving a character (usually) significantly younger (though sometimes older) than the person playing it. The most common ages to be ageplayed are birth through adolescence. While not a furry-specific term, it is an overlapping interest shared by many furs online and offline.
Ageplay in the furry community has taken on a bit of a different meaning for some. In the non-furry realm, ageplay is more specifically defined as a subset of BDSM. In the furry world, many who participate in ageplay are not into BDSM. The term more generally applies to anyone who simply plays a character younger than their player.
Ageplay is also commonly used to refer to the general interest of babyfurs, although in their case, it's generally within the ages of birth to pre-school. While technically the same definition, the mindset used when using the word, in babyfurs' own words, are under a different mindset. This has developed into conflicts at times between the different subset participants of ageplay.
 Ageplay vs. Pedosexuality
An important distinction to clarify: One need not be sexually attracted to the age of the character the player takes on. Some people who ageplay merely have a simple desire (whether it be enduced sexually or not) to relive that targetted age once again. Ageplay acts as a conduit for living out that fantasy. Commonly, when the desire is sexual, these people choose to interact with characters older than them, which would indicate their attraction for that age bracket, rather than their character's. In these cases, by definition, their urges are not pedosexual in nature.
Much of the talk regarding this type of roleplay has been heated, to say the least. Depending on whose side of the fence you're on, or whether you're walking the edge, everyone has a different opinion as to what's right and wrong, what's legal and what isn't. When the roleplay is sexual in nature, it's important to understand that while the subject matter of the roleplay is controversial, this type of roleplay is (most usually) performed by consenting adults. Such ageplay is sometimes mistaken for or equated to sexual relations with real minors, or the intent to perform such acts in real life, usually by those seeking to denigrate the ageplayer.
Of course (as in all legal matters), the age of consent in varying jurisdiction always applies. When roleplaying online via MUCKs, the laws where the server reside usually govern. Most MUCKs have created age-restriction policies for character registration based on the content and theme of the MUCK.
 Freedom of Expression
Many tough moral and legal battles have been fought in the name of both "sexual freedom", and "common decency". Players on both sides of the field have many valid arguments to their effect. Particularly within the U.S., it's the 1st Amendment which make such rights to sexual expression available to its citizens. Both the freedom to participate, or denounce it as an immoral practice, are protected by the same article of legislation.