Traditionally, bowling around the wicket is a tactic used to opposite handed batsmen, as the opposite is essentially bodyline bowling, though deliveries through this line are still occasionally employed to catch a batsman (particularly when unsettled) off guard.
The key change for a pace bowler is the angle of the delivery's trajectory. From a right arm seam bowler to a left handed batsman, the ball will angle across the batsman, towards the slips. Conventional outswing to a right handed batsman becomes inswing and the bowler often tries swinging in from this line at the stumps for an lbw decision.
When he comes around the wicket to the left hander, the angle crosses over the opposite direction. His arm releases the ball well wide of the wickets and so the ball is angling in to the batsman. This can be disruptive for many reasons. Primarily, it is the uncommon line along which the ball moves that affects batsmen. A ball which pitches at a width otherwise considered wide enough to cut can easily pass over middle stump. If the bowler swings or seams the ball, these deliveries are further confusing, as the batsman can become confused as to how best he should protect his stumps. In the batsman's favour, is that it is difficult to get out lbw to this, as the wide angle in means an on-target ball would usually hit well outside the line of off stump.
For spin bowlers, coming around the wicket can be a gold mine of opportunities. Again, predominantly to opposite handers, the use of a wide angle and turn can make the ball difficult to play.
A right arm leg spinner coming around the wicket to a left hander can be quite dangerous. If he hits the bowlers' footholes down the other end of the pitch (caused predominantly by right armed fast bowlers), not only does the ball carry the leg spin into the left hander, but it grips the pitch and turns in sharply. This, coupled with the natural angle in of his right arm (like a fast bowler coming around the wicket) can deceive a batsman quite effectively. Quite a number of batsmen have been given out lbw by not offering a shot to a ball bowled on this type of line. Defending balls of this nature can be awkward also, bringing bat-pad problems into play. Luckily for the left hander, such deliveries are prime for sweeping (the spin is your friend; hit against it and you will pay)
A left arm orthodox bowler can come around the wicket (often to both types of batsmen) to give his deliveries a straighter line. On a pitch that turns, this can tame the spin into something that a left hander will actually have to offer shots to. A good one can pitch on or outside his off stump and bowl him. To a right hander is more normal though. The left arm bowler will try to pitch the ball in line with the stumps for an lbw. He bowls the ball across his body with an angle, like a paceman coming around the wicket, but turns the ball back in, or in this instance, away from the right hander. Sharp turn can bring the ball well away from the stumps for a nick or outside edge, while more gentle turn will have the ball move straight on, which can be tough to predict and somewhat difficult to score off.