NOTE: The following is an undergraduate essay; it is not particularly rigorous and did not receive a particularly high mark when submitted. If you are really interested in Semiotics, I recommend checking other online resources such as Semiotics for Beginners
Saussure's theorising on la langue systematically omits so many types of sign-eg. the nonverbal and the natural-as well as the problems associated with a subjective consciousness as sign 'receiver', that it becomes next to useless as a method for studying communication. While a Saussurian semiotic might be employed in the examination of such notions as the fabled Inuit tongue's thirty different words for snow[1], it falls short of enabling any meaningful analysis of communication in general. Peirce's semiotic on the other hand encompasses a much broader range of signs, including language (in idealised "sound image" form, Saussure's langue, as well as the written form and the down-and-dirty everyday usage form, the actual spoken form, Saussure's parole), as well as nonverbal and natural signs.
While Saussure envisages the sign as an indivisible combination of sound image and idea, a dyad, Peirce considers the sign as a constituent part of an indivisible triad, along with the object and the interpretant. Where Saussure eschews the (understandably) popular notion that a sign is a thing which stands for something, because of the priority it accords to the object, Peirce accepts it, but with an important addition. Rather than a sign simply standing for something, a sign, according to Peirce, is a thing which stands for something to something. It is this third element, essentially incorporating the subject, which allows Peirce to live comfortably with the notion that a sign may represent an object.
Where Saussure avoids the problem of the subject by focusing on his idealised langue (in a somewhat rarefied paradigmatic and syntagmatic atmosphere), Peirce embraces the subject by introducing the notion of the interpretant, that being the effect produced by the sign in the mind of the interpreter (although Peirce does not actually consider consciousness prerequisite to an interpretant, in which case the interpretant takes the form of an action, or response), and including the concept of a ground, a basis or context with which an interpretant may relate an object and a sign. Hence Peirce shifts the focus of semiotics from an understanding of signs (and their relational value within a Saussurian system) to an understanding of semiosis, the process of sign interpretation.
Saussure's subdivisions of categories (linguistic and nonlinguistic, verbal and nonverbal, langue and parole, synchronic and diachronic, paradigmatic and the syntagmatic) are employed largely to remove anything which may complicate the systematisation of signs. Peirce's subdivisions serve not to reduce but to refine his system, excluding nothing, and distinguishing usefully between subcategories. One such subdivision (notably a trichotomy, in keeping with Peirce's penchant for three-ness) is applied to the sign-object relation, and helps to illustrate the inclusive nature of Peirce's model, as opposed to the more exclusive structuralist model.
Saussure insists on the arbitrariness of the sign, ie that the pairing of a sound image with a concept is dictated purely by convention, and not by any intrinsic relation between them. Peirce's model recognises such signs, and categorises them as symbols, along with other arbitrary nonverbal signs, such as a flag, or a siren. He goes much further however, recognising two other major categories of sign, in their relation to an object: 1. Icons, or signs which resemble their object in some way, like images, or sound effects (including Saussure's dreaded onomatopoeia), and 2. Indices, or signs which relate or refer to their object in some way, as smoke may signify fire, or a bruise may signify violence. In a Saussurian (structuralist) system, a sign functions as a link between the disparate universes of sounds and ideas, and it is largely the distribution of these links which is of interest to the semiotician. Within the Peircian model signs resemble, index or symbolise objects via an interpretant (which is formed in the context of a ground.) That is, a sign produces a response (an interpretant, which can be mental or physical) on the grounds of its relation to an object.
Unlike Saussure, Peirce embraces natural signs, such as a footprint, allowing that signification may take place outside of communication. In fact Peirce's interpretant does not even presuppose consciousness in the receiver of the sign; merely the capacity for response. Semiosis may be observed even in something as mindless as an automatic door: The disruption of an infra-red beam is a sign, whilst the door's response, to open, is the interpretant, the object being to "let a person enter". The ground for this door's interpretation is that people walking into the path of its sensor are likely to want to enter.
When a sign is experienced by a conscious mind however, the interpretant itself may in turn become a sign, thus leading to a new interpretant, and yet another sign. etc. This is "unlimited semiosis", and can be easily observed in conversation: An initial sign, such as a phone ringing, might lead to a person lifting the receiver and saying "hello". This is a verbal interpretant, but it will also quite clearly be regarded as a sign to the person calling that their call has been answered, leading to a new verbal interpretant, and a new sign, ad infinitum. Furthermore, unlimited semiosis can occur without communication, in the form of conscious thought. When we respond to a sign, our interpretant may be a thought, which itself becomes a sign, leading to a new thought. etc. Such an understanding of signs takes semiotics well beyond the scope of Saussure, allowing consciousness itself to be studied in terms of semiosis.
The fluidity afforded by the notion of the interpretant as sign (also effectively the object as sign, in that the notion of the object in the mind of the subject will bring about further associations, determined by the subject's ground) allows Peirce to describe a system which much more closely resembles the realm of real human communication. In Saussure's model, unlimited semiosis (or interpretation) sits nowhere comfortably, requiring a conceptual hopping back and forth between the two worlds of signifier and signified. Peirce's sign is part of an indefinitely large and complex structure, linking always to other signs, perpetually paraphrasing, and thus affording infinite variation in both dialogue and thought.
Peirce accepts the impossibility of signs existing independently of a subject to interpret them, and thus focuses his semiotic on the process of meaning-making rather than just meaning, which Saussure sought in the value of his dyadic signs through their relations to other signs within a larger system. It is the making of meaning which is most important in Peirce's model, where virtually everything may be regarded as a sign, capable of representing some thing to some one.
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[1] A popular idea which I was unable to verify, although a lesser known parallel is that the British have more than 70 different words for 'hill' - Alan Dawson, The Relative Hills of Britain, Cicerone Press, Milnthorpe, Cumbria, Ch. 7 - http://bubl.ac.uk/org/tacit/marilyns/chapter7.htm