The immune system is composed of cells. There are two arms to mammalian immunity - the innate immune system and the adaptive immune system. Each of these arms has distinct and unique cells that define their function, but each arm relies on the cells of the other to function. Innate immunity is found under the skin and mucus membranes and is evolutionarily conserved to recognise specific (yes specific) molecular components that classify distinct types of microbe and alert the adaptive immune system to bacterial, viral, yeast infections.
The adaptive immune system, alerted to infection by the innate immune cells then becomes armed. Adaptive immunity includes T lymphocytes and B lymphocytes which have on their surface receptors that were formed by gene shuffling of the component sections thus forming cells with monumental variation of their receptors. Some of these receptors would have recognised parts of our own bodies, but were destroyed during development. As the infection continues, and the microbes maybe mutate, the adaptive system can keep up by rearranging its B cell receptors. This process is called somatic hypermutation and is found only in B cells. The B cell receptors detach from the B cell and become the antibodies we rely on for immunity. T cell receptors will interact with infected cells and kill them, and they can also help activate B cells.
Yes the immune system is cellular. It has to be to enable it to monitor all possible points of entry of an infectious microbe. If it was a static organ like the liver or heart, it could not attack infection say in the foot. Being cellular enables the immune system to be mobile. The cells are transported in blood and lymph. T cells and to a lesser extent B cells circulate constantly throughout lymph and blood; entering checkpoints dotted along the lymphatic system called lymphnodes. It is within lymphnodes (static structures) that T cells and B cells will meet with innate immune cells carrying messages of infection.
I hope this helps a little.