This is an extremely complex topic outlined quite well here (1), and I will try to communicate this the best that I can. As with many things, brain function seems to change as a result of the interaction of two opposing processes, one is the way the brain deteriorates with age, and the other is the way in which it continues to remodel itself in order to compensate for that change. This response is going to deal with non-dementia and non-Alzheimer's related changes.
To really understand how this works, you need at least a basic understanding of neurogenesis, which is the process by which the body generates new neural cells from the relatively small amount of multipotent stem cells (2) in the nervous system. There is a review of this process here with its relationship to age. There is thought that non-coding RNA networks may play a role (3).
As far as changes which can be thought of as occurring due to the normal aging process, I'll generalize: There is a noted 10% loss of brain volume after age 80, mainly due to parenchymal shrinking. Also the normal aging process includes a reduction in rates of available neurotransmitters, and plaque accumulations, and neurofibrillary tangles can occur in the brain as it ages(these are much more likely in the Alzheimer's brain, but can occur as a result of normal aging). A more complete description of this process is available in the review here.
Studies of language aquisition in children have long been a benchmark for evaluatng this concept, with the common thought being it is much easier for children to learn languages than adults, however, this topic has its own complexities, and I'm going to stick to the more general concepts.
Even though our brains deteriorate with age, that is not all they do. We can use cerebral blood flow studies (4). to demonstrate that the brain does undergo a generalized age-related decline, but there is also evidence that activity increases in certain areas, possibly in an attempt to compensate for this decline.
Neural plasticity (5) is an interesting topic and neurogenesis does continue(6) up to the 80 year range, which seems to indicate we retain that ability to learn and retain what we've learned well on into that decade of life, here is one study which documents the regeneration of neurons in older age. However, the question of whether the ability of the brain is comparable to that of a younger brain is still being observed (7), and of particular debate is the age at which these degenerative changes begin to appear and become measurable, some state this may be as early as the late 20's, some state not until the 50's.
Many factors have been evaluated over the years for an association with this decline, some include the relation of exposure to low-level stress over the years, physical activity, hormone replacement therapy in women, and genetic factors. However, none have yielded any definitive answers.
Evaluation of this correlation was also attempted by evaluating neurological outcomes of children with brain injuries. However this study associated a younger age with a worse neurological outcome, seeming to indicate that the previous thinking that a younger brain is more malleable, therefore would do a better job recovering may not be exactly true. However, the counterargument could be made that injury at a critical state of development may produce a greater defecit than one slightly later, when connections have had more time to become solidified. And this concept seems to deal more with the brain's response to injury rather than natural development, but still bears consideration.
In short, I can not provide any definitive answer, only outline the current state of the science, which seems to allow for significant individual variation and the contribution of a variety of factors.
All numbered links will go to studies at pubmed. Other links will go to studies at medscape if you have access.