Pidgins are created when people who don't share a language try to communicate with each other. Pidgins do have grammar, of course, but they tend not to be carefully constructed; they're created on an ad hoc basis, to serve the needs of their speakers.
Many people have taken a more systematic approach to the creation of artificial languages. Some such languages (such as Quenya or Klingon) are created for fictional societies to speak; others (such as Esperanto) are intended to serve noble goals of international communication; still others (such as Loglan) are experiments designed to test certain claims or hypotheses; and many others are designed by hobbyists and linguists as an exercise or an entertainment.
Most such languages have their devotees. Enough Tolkien fans are interested in his invented Elvish languages Quenya and Sindarin (and the Tengwar and Cirth letters they're written in) to have formed the TolkLang mailing list; some people write poetry in various forms of Elvish. The sounds of the language are lovely. At the other end of the aesthetics scale is Klingon, created for Star Trek by Mark Okrand; it's a harsh, guttural language, suitable for warriors, and has attracted widespread public attention with the publication of a translation dictionary (and a project to translate the Bible and Shakespeare into Klingon, by the Klingon Language Institute).
Esperanto was created, in 1887, for very different reasons. Dr. L. L. Zamenhof intended his language (the name of which means "one who hopes"—named after Zamenhof's pseudonym "Dr. Esperanto") as an auxiliary language to facilitate communication among people who don't have a language in common—much like a pidgin, but with a more organized design. Most Esperanto words come from European-language roots, and letters are pronounced much like the letters of various European languages; these derivations and pronunciations, along with the streamlined structure of the language, make it relatively easy to learn for anyone who speaks a European language. (I've read that it's fairly easy to learn even for those who don't speak European languages.) Esperanto today boasts 100,000 speakers, and thousands of works of literature (both translated and written in Esperanto). There are magazines in Esperanto, a newsgroup, and an Online Esperanto course. For more information, see the multilingual Esperanto information page, which provides information in dozens of languages (including Esperanto).
And then there's Loglan. Loglan was created in the late 1950s by James Cooke Brown, for a very specific purpose: to test and explore the ramifications of the Sapir-Whorf hypothesis. (Loosely, that hypothesis is the notion that language determines thought. The theory, by the way, hasn't been proven or disproven (though I believe that it's been determined that Whorf's study of Native American time-oriented language and thought was flawed, but I may be wrong); my personal belief is that language and thought are complexly intertwingled and that they help determine each other.) More specifically, Brown's idea was that a language that minimized the limits it placed on thought would "release its speakers' minds from their ancient linguistic bonds." Loglan's vocabulary was designed to be "culturally neutral" (to the extent that's possible) and to be easily memorized by speakers of the "eight most widely spoken natural languages," and its syntax was designed to be logical (based on the Predicate Calculus) and free of ambiguity. Some loglanists see in Loglan the potential to be an ideal auxiliary language for international mutual understanding, based on its "clarity and lack of cultural bias." I'm skeptical, though I admit I haven't studied the language; I think pretty much everything has cultural bias of one form or another. (For example, the very notion that a language based on logic is a good idea shows a certain degree of cultural bias.) Also, loglanists believe that Loglan is ideally suited for machine translation and other computer-based uses, though I'm not sure how they address the difficulty of translating a natural language to and from Loglan by computer.
One interesting thing about Loglan (also true of Esperanto) is that its practitioners are aware of the necessity for language to grow and change to suit the needs of its speakers. Loglan's changes appear to be more formalized than those of most languages, but at least the language isn't static. Esperanto appears to be evolving through use, with changes being made informally by speakers, in much the same way that natural languages change. That kind of evolution has occurred to an even greater degree in American Sign Language (and other sign languages).
(I ought at some point to devote a column to ASL and other sign languages; for now, I'll merely note that ASL evolved from French Sign Language, developed by Charles-Michel, abbé de l'Epée, sometime in the 1700s. Over 500,000 people use ASL, ranking it fourth in number of speakers among languages in the US, and by far the most widespread of the invented languages discussed here.)
Lojban is one form that the evolution of Loglan has taken; it's apparently the result of an intellectual property dispute over the rights to Loglan. Lojban is a public-domain version of Loglan, with an entirely new set of vocabulary; according to the Lojban people, there are several hundred Lojban "supporters," and "over a dozen" people can speak the language, while Loglan (again according to the lojbanists) has more like a hundred supporters and "no known speakers."
If you're interested in constructed languages, you may be interested in joining the Conlang mailing list; see the FAQ for details. There's another mailing list, Auxlang, specifically for those constructed languages classified as "auxiliary languages," designed for international communication or cooperation. To subscribe, send "subscribe auxlang firstname lastname" to firstname.lastname@example.org.
Ailanto's language page contains pointers to many other interesting and related resources.
ui means "gladly" in Loglan. mi tavla do means "I talk to you" in Lojban.