Active immunity is happens due to the contact of lymphocytes with pathoges (or simply their antigens), which triggers the specific immune response and immunological memory is produced (T-Memory cells created). This response is slow and can take up to a couple days, whilst its effects are long lasting. The final products of the immunity are antibodies and T-Memory cells.
In passive immunity however, antibodies are taken into the body (could be from a poison antidote or from colostrum in human milk), and these antibodies only remain in the system for a short time, as then are then converted and excreted. This type of immunity is short term however it acts to clear the infection immediately. No immunological memory is created.
Both types of immunity can be artificial or natural.