Your keyboard contains a number of other characters, most of which are not properly punctuation marks at all, and very few of which are normally used in formal writing, except in certain specialist disciplines. Here are the ones which are found most commonly, or which can be produced with a word processor; such special symbols are often informally called dingbats:
Most computer keyboards lack the pound sign, but it can usually be produced in one way or another. If you absolutely can't produce a pound sign, it has become conventional in computing circles to use the hash mark instead (hence its other name):
The asterisk is occasionally used to mark footnotes. It also has one other rather curious use: it is sometimes used to replace a letter in writing a word which is felt to be too coarse to be written out in full, as in f**k. This is a usage mostly found in newspapers and magazines, in which writers are often careful to avoid offending their very broad readership. In most other types of writing, such words are normally written out in full if they are used at all. (Compare the somewhat similar use of the dash.)
A bullet may be used to mark each item in an enumeration if numbering of the items is not thought to be necessary; look at the summaries at the ends of most of the earlier sections of this document.
The at sign is chiefly confined to business documents, in which it stands for `at a price of ... each':
The paragraph mark and the section mark are occasionally used to represent the words `paragraph' and `section' when referring to some part of a work: in ¶ 2, in § 3.1. They are only appropriate when brevity is important, such as in footnotes; in your text, you should write these words out: in paragraph two, in section 3.1.
The remaining symbols in my list have various particular uses in specialist disciplines, and sometimes in dictionaries, but they have no function in ordinary writing.