Supplement

[1] We make use of signs to refer to something that is not directly present. A sign replaces something else. It is a substitute, a representative of something other than itself. A sign is a supplement, a supplemental aid that can be deployed when the presence itself falls short. (A score can be regarded as a supplement. A consideration of this idea can be found on the page entitled No (-) Music - D. Schnebel.)
The sign evokes the ideal of a presence without signs, a direct contact with what is represented by the sign. It is considered an imperfect and incomplete surrogate that is expected to substitute for something that is only temporarily absent. It is an exteriority, an outside, a material signifier that is added afterwards to the inner integrity of a signified. The signifier only represents the signified. Unfortunately, however, something of the original richness of the signified always is lost in its representation. That is why the supplement is preferably considered as inessential, a non-required surplus that really should not need to be added to the pure fulness of the interior. 'What is added is nothing because it is added to a full presence to which it is exterior. Speech comes to be added to intuitive presence (of the entity, of essence, of the eidos, of ousia, and so forth); writing comes to be added to living self-present speech' (Of Grammatology, p.167). The sign as a supplement is thus in a hierarchically subordinate position to the essential or the present.

[2] However, a sign is not merely an subordinate substitute. This is shown in the double meaning that may be assigned to the term supplement. Supplement means both replacement and addition, that which at once supplements and supplants. It can mean the adding of something to something that is already complete in itself; or it can mean the adding of something to complete something (cf. Of Grammatology, p.144-5). And the shadow presence of the other meaning is always there to undermine the distinction. This 'dangerous supplement' (Rousseau) is therefore an undecidable, an unsettled concept.
A supplement is a substitute for something else that is unable to be present. Supplements are called in precisely because there is a lack in what is supplemented. However, a supplement also adds something; it is a surplus. (If a supplement is both added to something and replaces that same thing, it means that they are neither strictly opposed to one another, nor equivalent to each other.)
If a presence needs a sign in order to appear, then it follows that it is always already permeated from the inside by a shortage, a lack. If it cannot do without the supplement, then the supplement becomes the precondition for the presence of the present (cf. Of Grammatology, p.184). The originality of the signified is only afterwards produced with and within the supplement; it appears only afterwards in and through the sign. The sign therefore becomes a prerequisite for the signified to appear. At the same time, however, the sign postpones a direct contact with the origin, with that what is represented, both temporarily and spatially. On one hand, the supplement is an exteriority, outside of the signified and outside of itself because in its quality of being a sign it represents something different than itself. On the other hand, however, a supplement enables an interiority and produces the inside. Every supplement postpones that which it installs. And every signified is, in fact, an effect, a trace of a signifier.
(Derrida's point of interest is the difference between the represented and that which represents. Something remains hidden in every representation, something is forgotten, suppressed, or excluded. This  could be called 'the other'. Every perspective that makes use of language fails to cover all aspects and to address all matters concerned; this is what Derrida calls to our attention.)

[3] Writing is the supplement of thought. Writing should serve as the vehicle to carry thought.
The first problem is that thought cannot appear outside writing. This, in fact, means that everything starts with supplementation. Without this supplementation there would only be silence. Something at the core of thinking seems to be missing. Thought needs the supplement of writing in order to be whole, and thus thought without writing is not (fully) itself.
The second problem is that writing, instead of serving as a transparant medium to carry thought adds itself to thought and then substitutes itself for thought. What should be merely a means of expression affects or infects the meaning it is supposed to represent. What began as a supplement to help thought to present itself becomes a replacement that threatens the integrity of what it intends to replace. Thought cannot exist without writing. A signified cannot appear without a signifier. This means that a supplement cannot simply represent the absent signified. Because the signified can only be(come) present through a signifier, each signifier can only be substituted for another signifier that maintains another relation with the deficient presence. 'The supplement is always the supplement of a supplement. One wishes to go back from the supplement to the source: one must recognize that there is a supplement at the source' (Of Grammatology, p.304). A process of substitution of supplements is thus started to which there is no beginning and no end.

[4] Plato's Supplements offers an example of the working of a supplement in a text by Derrida. In Teaching a Supplement, the double meaning of supplement is related to the teaching of jazz music.