AS = Autonomous System, a network of one or more (usually more) computers that looks like one entity to the outside Internet. It's 'autonomous' in that it can route traffic within itself without help from any outside source. The Internet is, at a high level, a collection of ASes that all pass data among each other. Every AS has a globally unique number, usually represented as AS15169 for AS number 15169.

BGP = Border Gateway Protocol, a specific Exterior Gateway Protocol that allows ASes to figure out what other ASes are close by and to which of their neighbors they should route traffic destined for a specific IP address. This basically works by each AS advertising which groups of IP addresses (represented by prefixes) they know how to reach. A prefix is something like 10.0.0.0/24, which represents all addresses from 10.0.0.0 to 10.0.0.255; in a prefix, the number after the slash is how many bits of the IP address are fixed. In a /24, 24 bits, or three eight-bit bytes, are fixed, so the last eight bits can vary freely. Larger numbers indicate smaller blocks of addresses, unintuitively enough. For example, AS15169 advertises that it contains 173.194.0.0/16, or the range 173.194.0.0 - 173.194.255.255. Route advertisements contain cost information, which is primarily due to how long the path is; as an example, if I'm AS1 and I contain the range 10.0.0.0/24, I'll advertise that with a very low cost. If I hear from my neighbor AS3 that she contains 192.5.0.0/16, I'll advertise that with a higher cost, since I'll have to hand it off to a different AS.

(Edited to add: Apparently, IPv6 prefixes work essentially the same as IPv4 prefixes. http://www.sabi.co.uk/Notes/swIPv6Prefixes.html )