A prefix is an IP subnet, basically.

Most here are familiar with Google's public DNS resolver, 8.8.8.8, so let's use that as an example.

Google has been assigned the autonomous system number (ASN) 15169. The 8.0.0.0/8 IP address space was allocated to Level3 and they have reallocated a small part of it, 8.8.8.0/24, to Google.

Google, when speaking to their BGP peers, "announces" routes, including an announcement for the 8.8.8.0/24 prefix.

An announcement is basically Google's router telling, for example, Level3's router, "Hey, I know how to get to 8.8.8.0/24. If you have traffic going there, you can send it to me."

Level3 then passes that along to other peers, like me. Level3 says, "If you have traffic for 8.8.8.0/24, you can send it to me and I'll send it on towards its origin". The announcements continue to be passed along to other peers.

A withdrawal is the opposite of an announcement: "Hey, I don't have a route to 8.8.8.0/24 anymore" or, in many cases, "the route to 8.8.8.0/24 has changed, here's the new one".

Here's a (slimmed down) example from one of my routers:

  # sh ip bgp 8.8.8.0/24 | b 3356
    3356 15169, (received & used)
      4.69.248.15 from 4.69.248.15 (4.69.180.167)
        Origin IGP, metric 0, localpref 100, valid, external, best
        Community: 3356:3 3356:86 3356:575 3356:666 3356:2042

This shows the "AS path" (the path through the various ASNs back to the origin), "3356 15169". AS3356 is Level3. We can see that AS15169 (Google) originated the prefix (8.8.8.0/24), and announced it to their BGP peer, Level3. Level3 then passed that announcement along to my router.

Here's the same thing for 66.232.224.0/24, mentioned in the article:

  # sh ip bgp 66.232.224.0/24
  BGP routing table entry for 66.232.224.0/24, version 1125190207
  Paths: (1 available, best #1, table Default-IP-Routing-Table)
    Advertised to update-groups:
          4
    3356 209 40839, (received & used)
      4.69.248.15 from 4.69.248.15 (4.69.180.167)
        Origin IGP, metric 0, localpref 100, valid, external, best
        Community: 209:209 209:13070 3356:3 3356:22 3356:86 3356:575 3356:666 3356:2042

The AS path here is "3356 209 40839" which shows that traffic from me to 66.232.224.0/24 will first go to Level3 (AS3356), then to Qwest/CenturyLink (AS209), and finally to AS40839 (Kohl's Department Stores).

With regard to the hijack described in the article, this means that another organization announced the prefixes into BGP, effectively saying, "So, all of that traffic you have that's going to Apple/Facebook/Google/etc., just go ahead and start sending that to AS39523 now". Instead of ending up at Apple/Facebook/Google, it'll instead be redirected to this unknown organization in Russia.

(N.B.: Before anyone chimes in to "correct" something, this is a bit simplified. There's much more to BGP than this and there are a number of different factors which determine which route is chosen. A longer prefix -- mentioned in the article -- is one way to "defeat" a better route.)