As humans, we process language very easily many times a day in our normal communications with each-other. Hence, in our terminology, we are able to compute a meaning for a given utterance (since we understand each-other), and to compute an utterance for a given meaning (since we can express our thoughts). We want to know how humans perform this task, in order to further our understanding of natural language. In this sense, computational linguistics is part of linguistics and cognitive science.
A second motivation to study computational linguistics is of a more practical nature. If we know how to compute this relation between form and meaning, then we can write computer programs which perform this computation. Such computer programs will make a broad set of interesting natural language applications possible: spoken information systems, machine translation systems, natural language interfaces, and many others. One of the results of the NWO Priority Programme on Language and Speech Technology is a spoken dialogue system for public transport information. The system is accessible by telephone. A caller can request (in ordinary Dutch) time-table information for all Dutch train connections. The system operates automatically: if all goes well, no human interaction is required. In such a system the computer analyses the utterances of a user in order to find out what connection is being requested. Furthermore, natural language synthesis is used to produce further questions (for instance if not all information is available for a database lookup yet), and to produce the resulting connection, once the database has been consulted. From this perspective, computational linguistics is an engineering science (the term human language technology is sometimes used for this type of work).
The third motivation to study computational linguistics is theoretical: we are interested in the computation of the relation between form and meaning for its own sake. This computation has interesting formal properties and relates in interesting ways to theoretical aspects of the theory of computing in general. Construed in this way, computational linguistics is closely related to mathematical linguistics and theoretical computer science.