Here, Microsoft uses the names "concurrent" and "background" to describe two versions of the GC it uses in .NET. In the .NET world, the "background collector" is an enhancement over the "concurrent collector" in that it has less restrictions on what application threads can do while the collector is running.
A basic GC uses a "stop-the-world" strategy: applicative threads allocate memory blocks from a common heap. When the GC must run (e.g. too many blocks have been allocated, some cleanup is needed), all applicative (managed) threads stop. The last stopping thread runs the GC, and unblocks all the other threads when it has finished. A stop-the-world GC is simple to implement but induces pauses which can be perceptible at the user level.
Microsoft's "concurrent GC" is generational: it uses the stop-the-world strategy for only a limited part of the heap (what they call "generations 0 and 1"). Since that part remains small, pauses remain short (e.g. below 50ms), so that the user will not notice them. The rest of the heap is collected with a dedicated GC thread, which can run concurrently with the applicative threads (hence the name).
The concurrent GC has some limitations. Namely, there are moments when the GC thread must assume a somewhat exclusive control of the heap. During such times, applicative threads may allocate blocks only from small thread-specific areas. Threads which have bigger needs will soon stumble upon the main heap, which, at that time, is locked by the GC thread. The allocating thread must then block until the GC thread has finished its lock-the-heap phase. This again induces pauses. Less pauses than with a stop-the-world GC, and these pauses do not affect all threads. Yet pauses nonetheless.
The "background GC" is an enhanced GC in which the GC thread needs not lock the heap. This removes the extra pauses described in the previous paragraph; only remain the limited pauses when the young generations are collected (what Microsoft calls "a foreground collection").
Note: there are "hidden costs" with the concurrent GC and the background GC. For these GC to operate properly, memory accesses from applicative threads must be done in some very specific ways, which have a slight impact on performance. Also, the GC thread may have an adverse effect on cache memory, thus indirectly degrading performance. For a purely computational task with no need for user interaction, a stop-the-world collector may, on average, yield somewhat better performance (e.g. a twenty-hours-long computation will complete in nineteen hours). But this is an edge case, and in most situations the concurrent and background GC are better.