Cytokines are essentially signalling molecules of the immune system. Broadly speaking these can be classed as pro- or anti-inflammatory. Pro-inflammatory cytokines promote inflammation, and anti-inflammatory cytokines inhibit inflammation.
Inflammation (again, broadly speaking) is associated with the 'innate' immune response - this is the immediate response by neutrophils and macrophages (among others), and these cells release a variety of molecules designed to damage cells around them. This is great if those cells are bacterial, or damaged host cells that need removing, but not so great if that is 'normally' functioning tissue.
Aging is associated with increased systemic inflammation - i.e. a greater proportion of pro- to anti-inflammatory cytokines. The causes of this aren't entirely certain, but it could be that the adaptive immune response (i.e. the memory T-/B- cells) is not as effective in older ages so the innate response is more important, or that it is just simply left 'unchecked'. Either way, the result this that the pro-inflammatory response can end up damaging host tissues, including the brain, without the corresponding repair instigated by anti-inflammatory cells/cytokines.
Any anti-inflammatory medication (e.g. aspirin, ibuprofen, or glucocorticoids) will, in theory, reduce inflammation, but by suppressing the immune system the risk of infection naturally rises, and there may be unforeseen consequences to prolonged use (the immune system does a lot more than just fend of bacteria - e.g. it is essential for tissue maintenance and regeneration), so I am by no means suggesting the use of anti-inflammatories to protect brain function!