Sequence motifs are conserved sub-sequences which have structural or functional significance. In DNA, they could be promoters/terminators, enhancers/silencers, transcription factor/enzyme recognition sites. In RNA, they could be riboswitches, and in proteins, these could be nucleotide-binding domains.
Sequence motifs come in three different flavours:
1. Palindromic motifs
These types of motifs are also palindromes. An example of this is 5'-GCATATGC-3' because 5'-ATGC-3' is the reverse complement of 5'-GCAT-3'.
2. Gapped motifs
Also known as spaced or dyad motifs, these exist in two parts, with a fixed-length non-conserved gap in between.
3. Degenerate motifs
These take into account SNPs. For example, 5'-TTATMCAMA-3' is a DnaA box which occurs near + within the origin of replication in E. coli. The fifth position is known as degenerate because M = C or A. Note that degenerate motifs can also be gapped or palindromic, or both!
4. Regular motifs
The remainder of motifs fall in to this category.
Some parts of this post are based on the following paper:
Das, M.K. and Dai, H.K., 2007. A survey of DNA motif finding algorithms. BMC bioinformatics, 8(7), p.S21.