(This article is part two of a three-part series on inflammation. Find part one here)
The Brain And The Immune System
Until about 10 years ago, the prevailing school of thought was that our immune systems and our brains were mutually exclusive entities.
There's a membrane separating the blood of the body from the blood in the brain (called the blood-brain barrier). The cytokines and other signaling molecules that regulate the immune system are much too large to cross through the tiny holes of the membrane. Thus, brain and immune must operate separately - or so was the thinking at the time.
Turns out, this is not exactly the case. While cytokines and other immune signaling molecules are indeed too large to pass through the blood-brain barrier passively, researchers have discovered that there is a mechanism for active transport of cytokines into the brain. Furthermore, we've learned that even when cytokines don't cross the blood-brain barrier, they can still transmit a signal across the barrier, thus resulting in increased cytokine production in the brain. 
So now we know: the brain and the immune system are NOT separate entities. If the immune system triggers an inflammatory response in the body, there will also be some degree of neurological inflammation.
The Effects Of Inflammation On The Brain
Once researchers learned that the immune system could indeed initiate an inflammatory response in the brain, it opened up countless new doors for models that involve inflammation to explain certain neurological conditions. Over the past decade, we've learned a ton about how neurological inflammation effects certain cognitive and emotional functions.
Brain Fog - "Brain Fog" is obviously not a strict scientific term, but it is useful because it perhaps best describes the way you would experience neurological inflammation. We all know the feeling; your brain feels slow, attention is scattered and high-level connections between concepts simply aren't happening. Your brain gets tired easily and memory formation and recall are poor. Unfortunately, a lot of people have become accustomed to systemic inflammation accepted this low-functioning mental state as baseline.
In reality, "brain fog" is a the resulting amalgam of the other effects listed below, in combination with the down-regulation of energy metabolism caused by inflammation we mentioned in the previous article. Remember, the brain consumes somewhere around 25% of the body's energy over the course of a day (despite accounting for only about 2% of body weight). When energy metabolism is reduced, the brain is the organ that suffers the most.
Memory Impairment - When the brain is inflamed, memory formation and recall both suffer. The greatest amount of research here had been done with elderly individuals, but there's plenty of evidence to suggest that younger brains also show memory impairment when neurological inflammation is present. 
Depression - This is part of the new paradigm of research around depression. For the past 30 years-or-so, researchers have looked at depression as fundamentally a neurotransmitter-imbalance problem. Newer evidence suggests that neurotransmitter imbalances may only be a symptom of a larger issue. Depression seems to be a brain-structure issue at it's core.
Unsurprisingly, inflammation plays a big role in the physical/structural changes in the brain that lead to depression.
This may be another example of an evolutionarily-derived response gone haywire. As I mentioned in the previous article, the immune system down-regulates energy metabolism and pain tolerance during an inflammatory response to discourage the animal (in this case, us) from going out and foraging in a time of vulnerability. It now seems that depression in the individual might be part of the same response.
Regardless, what the evolutionary reason is, neurological inflammation has recently been closely correlated with depression. These studies are relatively recent, so I expect knowledge here will expand quite a bit in the coming years. 
- Cognitive Decline and Alzheimer's Disease - If you are approaching retirement age, neurological inflammation is of particular concern for you. The most recent research on age-related cognitive decline and Alzheimer's Disease suggests that neurological inflammation may be the primary driver in the progression of these diseases. Older individuals with elevated levels of inflammatory cytokines were several times more likely to develop symptoms of cognitive impairment.