J.R.R. Tolkien called it "A Secret Vice". But in these days of widespread Internet popularity, the hobby of language building ("conlanging" as it is frequently called) is beginning to be a bit less secret than it was in Tolkien's time. What inspires someone to begin the challenging task of designing a language? There are as many reasons for the hobby as there are hobbyists. It can begin with many small steps and proceed through more and more elaborate language sketches. The exact point where simple linguistic doodling turns into something that can reasonably be called a language is hard to pinpoint. But there are a few characteristics shared by all natural languages that are required before the stage of "language" can be reached.
All spoken languages have a finite set of sounds that can be used in certain combinations to form meaningful units of speech. These sounds themselves, the "phonemes" as they are called, have no meaning, but they must be used in the context of other speech sounds. Even non-spoken languages have analogous units of arbitrary strokes or gestures that combine to form meaningful signs and symbols. If you compare the sound systems of various languages, you will find that different languages divide the range of speech sounds in different ways, and no language uses all possible human speech sounds. The first step in creating a language is to decide which sounds will be used to distinguish minimal contrasts between words (the phonemes of the language), which combinations of phonemes are allowed to occur (phonotactics), and how the phonemes actually sound in various contexts. In the course of designing the language, the phonology may change, but it makes things easier to decide on a general scheme for the phonology early on (like painting the background of a picture).
Most languages have some means of building words from smaller meaningful units called "morphemes", although a few languages of the "isolating" type have only a single morpheme per word. In general, the morphology of a language ranges from agglutinative (morphemes simply joined together one after the other to form a word) to fusional (two or more morphemes combining into a third, often unrelated form representing the combined meaning of both), and from isolating (a single morpheme per word) to polysynthetic (a large number of morphemes in a single word, incorporating parts of the sentence that other languages would use separate words for, such as a verb with its subject and object). Morphology describes how parts of words are put together, for instance by adding prefixes and suffixes to a root, changing a vowel or a tone here and there, or combining multiple roots into a single compound word. Different categories of words (such as nouns, verbs, and adjectives in English) have their own separate rules for morphology.
Finally, all languages have rules describing the organization of words into phrases and sentences. Some languages place the verb at the beginning of a phrase, others at the end, and still others (like English) in the middle. Languages also differ on the placement of modifiers (such as adjectives) before or after the words they modify; English for instance has adjectives typically preceding nouns, but French more often puts them after the nouns. English puts relative clauses after nouns, but Japanese puts them before. Different languages have different ways of joining phrases and clauses. These kinds of rules are easily overlooked, and it tends to be only the more fully developed languages that have well-defined rules for syntax. But it's important to avoid falling into the habit of imitating English word order, unless of course you're intending your language to be a new variety of English.
Generating vocabulary is the most time-consuming part of language design. While natural languages may have thousands of words, each with a complex interrelated set of meanings, this is an aspect of language in which artificial languages tend to be more limited. As a first step, there's nothing wrong with defining a word with an equivalent in English or another familiar language. But there will inevitably come a time when one or more English connotations seems inappropriate for the new language, and the definition will need to be refined. In certain well-defined semantic areas, such as words for colors, it might be advantageous to start from scratch without reference to English. Experience with a variety of natural languages can also be helpful in avoiding some peculiarities of English vocabulary.
Putting it all together
Phonology, morphology, syntax, and semantics are the basic components of all languages. The trick of language creation is to fuse these elements into a coherent, useful, and appealing whole. This synthesis of language elements can only be tested by writing text in the language, either original text or translations from other languages.
Here are a few books that might be helpful in designing a language.
Peter Ladefoged & Ian Maddieson, The Sounds of the World's Languages (Blackwell Publishers Inc, 1996) ISBN 0-631-19814-8 (hardback), ISBN 0-631-19815-6 (paperback)
Handbook of the International Phonetic Association (Cambridge University Press, 1999) ISBN 0-521-65236-7 (hardback), ISBN 0-521-63751-1 (paperback)
Peter T. Daniels & William Bright (editors), The World's Writing Systems (Oxford University Press, 1996) ISBN 0-19-507993-0 (hardback)
Thomas E. Payne, Describing Morphosyntax (Cambridge University Press, 1997) ISBN 0-521-58224-5 (hardback), ISBN 0-521-58805-7 (paperback)