Peptides are short chains of amino acids that make up long-chain proteins.  There is no specific definition for the naming of peptides, though the PCPC (Personal Care Products Council) defines peptides containing 2-10 amino acids as small molecule synthetic peptide, those containing 11-100 amino acids as oligopeptide, and those containing more than 100 amino acids as polypeptide.  In the 16th edition of INCI (International Nomenclature Cosmetic Ingredient) dictionary, synthetic peptides consisting of more than 100 amino acids are named as "polypeptide".  However, when we say "peptide", we usually refer to small molecule synthetic peptides.
Generally speaking, monopeptides, tripeptides and pentapeptides can help promote collagen production;  hexapeptides can block the release of neurotransmitters, which is similar to the efficacy of the botulinum toxin[1].

Divided into the following by efficacy
Anti-aging peptide: palmitoyl tripeptide-5, palmitoyl pentapeptide-4, tripeptide-1 copper, pentapeptide-3, acetyl hexapeptide-8, palmitoyl tripeptide-1, etc.

Soothing peptide: palmitoyl tripeptide-8, etc.

Lightening peptide: acetyl tetrapeptide-2, etc.

Be classified by their mechanism of action
Signal peptide[2]: such peptides can stimulate the production of extracellular matrix proteins (including collagen and elastin) and reduce the activity of collagenase, improving skin aging;

Carrier peptide[2]: this peptide can stabilize and deliver metal ions. Taking blue-copper peptides as an example, when copper ions enters the skin, it promotes collagen production and exerts anti-aging effects;

Neurotransmitter inhibitory peptide[2]: it acts as a blocker of neurotransmitter release present in the junctional vesicles to reduce subconscious muscle movements and power facial fine lines and wrinkles, such as the commonly seen Argireline.

References
[1] Du DX, Wang SS, Zhou L. Application of peptides in cosmetics[J]. Journal of Qilu University of Technology (Natural Science Edition), 2012, 26(001):35-39 (in Chinese).

[2] Zone Dina Draelos MD, et al. Cosmeceuticals [J]. People's Military Medical Press, 2007.