Senior in the USA refers to the fourth year of a standard four-year college degree (an undergraduate degree or BA, for most Commonwealth English speakers). Students in the four years of a standard US college degree are known respectively as freshmen, sophomores, juniors, and seniors.
(Confusingly, the last two years of high school — roughly, ages 17–19 — are also known as junior and senior year. This is usually disambiguated by context, though.)
Rising senior genrally means that the person in queston is in between designations, but that senior will be the next applicable one. If I was a junior in the 2010–2011 academic year and will be a senior in the 2011–2012 academic year, then right now (summer 2011), I am a rising senior. It’s similar to the usage of going on in a phrase like “Jessica is six and a half, going on seven.”
If I’m right in reading this as a US or possibly Canadian usage, then it’s quite unrelated to the use of senior in phrases like senior common room; this is just one of those times when transatlantic differences really start to get jarring. I’m sorry…