Paradigm: A paradigm is a set of associated signifiers (the form) which are all members of some defining category, but in which each signifier is significantly different. In natural language there are grammatical paradigms such as verbs or nouns. In a given context, one member of the paradigm set is structurally replaceable with another. The use of one signifier (e.g. a particular word or a garment) rather than another from the same paradigm set (e.g. adjectives or hats) shapes the preferred meaning of a text. Paradigmatic relations are the oppositions and contrasts between the signifiers that belong to the same paradigm set from which those used in the text were drawn.
So a paradigm is ‘a set of associated signifiers which are all members of some defining category’. So cat could fall into the category animals.
All of the above from: http://www.aber.ac.uk/media/Documents/S4B/sem-gloss.html#P
Syntagm: A syntagm is an orderly combination of interacting signifiers which forms a meaningful whole (sometimes called a ‘chain’). In language, a sentence, for instance, is a syntagm of words. Syntagmatic relations are the various ways in which constituent units within the same text may be structurally related to each other. A signifier enters into syntagmatic relations with other signifiers of the same structural level within the same text. Syntagmatic relationships exist both between signifiers and between signifieds. Relationships between signifiers can be either sequential (e.g. in film and television narrative sequences), or spatial (e.g. montage in posters and photographs). Relationships between signifieds are conceptual relationships (such as argument). Syntagms are created by the linking of signifiers from paradigm sets which are chosen on the basis of whether they are conventionally regarded as appropriate or may be required by some syntactic rule system (e.g. grammar).
The above from: http://www.aber.ac.uk/media/Documents/S4B/sem-gloss.html#S
The above from: http://tinyurl.com/kus6f5x
So I think the best way to describe a syntagm is ‘an orderly combination of interacting signifiers which forms a meaningful whole’.