Beatgrids are useful because they mark where the downbeat is in a track and after telling Traktor the tempo of a track, Traktor creates a grid of the track in order to match the beats to another song's grid. You can also put in multiple beatgrids which is useful if a song changes tempo in the middle or if it's a vinyl rip and has a drift of the tempo.
In order to a Beatgrid marker, you find the first downbeat and line up the cue marker on it. You can use Snap, Vinyl, or CD mode, it's your preference.

Once you have found the downbeat you select the drop down in the cue box and select Grid.

This will now have the tracks beats marked out.

It's usually a good idea to jump to near the end of the track and see if the lines still match up with the beats.

In this case they do, other times you may need to tweak the track's tempo in the BPM panel until you have them line up, you can also turn on the tick / metronome and see if that matches up with the beats.
For a more in depth article on beatgridding, see Ean Golden's article on DJTechtools.com.
Cue points are much like beatgrid markers but they are used just for marking points in a track that you may want to go back to.
To mark a point in a track as a cue point, you just need to press the "Lock" button in the Cue panel, this will drop a cue marker and lock it there so even if you scroll ahead or behind you still have a cue marker there.
These come in handy for jumping through a track like if you are doing a live remix where you want to jump to a break and then go back to a build up or jump back to the chorus one more time before the song ends.
An even more fun part of Traktor is setting loop points. To set a loop point you have a few options. You can do it live while the track is playing by pressing Set/In in the Loop Set panel to set the start of the loop and then press Active for mark the end point.

Or you can press the preset loop button in the actual transport panel.

It really depends on personal preference. You can change the loop size by either pressing the sizes in the loop set box or using the Loop Start and Loop Start panels. You can also move loops by using the Loop Move panel. It is good to note that you can do this all live while a track is playing.

If you would like to save a loop so that it is still there when you bring the track up again you press the Lock button in either the Loop Set panel or the Loop Select panel. Then the next time the track is playing, before or while it is in between the Loop markers, you just press the Active button in the Loop Set panel. To come out of the loop you just press Active again. After you leave the loop, you can jump right back to the In marker by just pressing Reloop.
That pretty much wraps up getting started using some of tools for creating live remixes. Some resources to check out for more info on Traktor would be djtechtools.com and also the Traktor Manual and Traktor Tutorial DVD.
Also, if you have Ext. + Stripe selected in your Collection preferences and you save your ID3 tags to your track, it will write your cue points and loop markers to the ID3 tags so you have them even on another computer.