Quick intro:
Hypersensitivity reactions occur due to the introduction of a certain antigen (Ag) which the body finds foreign and thus tries to eliminate it with the help of certain cells and antibodies (Ab). This conflict produces an immune process, a war-like phenomenon known as hypersensitivity reaction.

Remember the mnemonic: [ ACID ]:

Type I; Anaphylactic/Atopic
- Mast cell or basophil + Ag + IgE
- e.g. Anaphylaxis (e.g. to penicillin), Asthma

Type II; Cytotoxic (kills cells)
- Affected cell + Ag + IgM/IgG
- e.g. Autoimmune hemolytic anemia, Rh factor dse, Rheumatic fever

Type III; Immune complex
- Ag + Ab + complement (e.g. C3, C5)
- e.g. Post-streptococcal glomerulonephritis, SLE, Rheumatoid arthritis

Type IV; Delayed (T-cell mediated)
- T lymphocytes + Ag » formation of lymphokines
- e.g. TB skin test, Transplant rejection, Contact dermatitis

Some information, including the mnemonic, was taken from First Aid for USMLE Step 1 book.