The complement system complements or enhances the immune system. These proteins cause a series of chain events that induce inflammation that  helps antibodies clear infection.  There are three pathways by which complement helps the immune system:

The classical, alternative, and lectin pathways.

The complement system and disease

The complement system is a perfectly normal part of the immune system but sometimes it goes haywire.  Systemic lupus erythematous is a famous example.  But there are also some rare diseases such as Degos/malignant atrophic papulosis.

Here is an excellent YouTube video explained the complement system presented by MEDSimplified

References

https://www.ncbi.nlm.nih.gov/books/NBK27100/