There are a variety of cells involved in immune responses. The major important ones are:
1) phagocytes (such as macrophages and neutrophils); these cells ingest other harmful cells and break them down sort of like how you ingest and break down food. They are usually present on site to react immediately to infection but they have their limits and once they have consumed a certain number of invaders they die. Some of them can also be directed to consume particular entities using antibodies.
2) helper T cells; these cells do exactly what their name implies, they help. They don't diretly do any killing, what they do is they release chemicals that act like chum for real killing immune cells, attracting them and intensifying the immune response when the helper's detect the presence of an invader. Think of them as immune response cheerleaders.
3) killer T cells; these cells are really mean buggers, they possess some very potent chemicals within them that disintegrate other cells along with a set of protein complexes that function a lot like a hypodermic needle able to penetrate through the outer coatings of harmful pathogens. Killer T cells will bind to a pathogen and drill through the outer coating in order to release the potent chemicals into that pathogen and destroy it. One of the interesting things about Killer T's is that they are the cells in your immune system best equipped to deal with some of the more difficult pathogens to combat such as viruses and cancers.
4) B cells; these cells are responsible for the production of antibodies once they are activated as (they then become known as plasma cells). For pretty much each pathogen there is a specific antibody produced that responds to that pathogen and only that pathogen. Antibodies bind to that pathogen and in doing so cover up its active sites (the areas on the pathogen that allow it to bind to, and interact with, your body through which it performs its harmful effects) thereby neutralising the pathogen. The antibodies also act as big "eat me" flags for phagocytes, attracting them and encouraging them to consume the antibody-coated invader.
5_ Memory cells; whe you encounter a pathogen for the first time not all immune cells engage in the fight, some become dormant and sit in places like your lymph nodes (read up on Wikipedia about what the lymph system is but it is sort of an overflow system for the liquid component of your blood among other things). Memory cells are these dormant cells. They live a very long time and, having been created in response to a specific pathogen, carry the knowledge of how to make specific antibodies dealing with that particular pathogen. When the memory cells again detects the pathogen later it allows your body to respond far more quickly to that pathogen through the production of new B cells already tailored specifically to fight that pathogen.
Depending on the type of pathogen present your response will be different. Antibodies for example are not effective against cancers while killer T's are somewhat effective in some circumstances (though often killer T's) aren't enough and you require medical intervention such as chemo and radiotherapy). Antibodies are great against bacterial infections for example. There are also many different types of antibodies and your body cycles through each type in turn as longa s the infection endures in an attempt to find an effective type.
Note I have simplified this a little so some of what I have said is not strictly true but will do for the purposes of a basic explanation.