There's several common repairs.
It all depends on the type of damage, size of damage and the type of carpet. It also depends on the customer's expectations & requirements. Most of all it depends on the economics of the situation, i.e. you wouldn't spend £200 on repairing a 5 year old carpet that cost £300.
The most common way to sort out small things like cigarette burns is to retuft/reweave. This involves pulling out the affected tufts and replacing them with fresh ones which can either be taken from an off-cut if available, or by pulling up the carpet at the edge and nicking a few from there.
On a tufted carpet you break up the latex between the primary and secondary backing when you do this, so I help to secure the new tufts in place by injecting a small amount of latex adhesive into the carpet with a hypodermic syringe, to stick the layers back together again. You have to be very careful doing this, not to get the adhesive any higher than the base of the tufts, otherwise you can make a right mess of it!
On woven carpets it's a doddle, easier than tufted IMO. You just have to re-thread tufts in and that's that, the backing holds them quite securely. I'd maybe add a bit of latex if it was in a traffic area, just to be on the safe side.
After retufting, the area should be combed up and trimmed with napping shears to level off the new tufts.
There is a problem with this sometimes. If the new tufts are a lot cleaner than the old tufts it will show up! Obviously cleaning the repair area will lessen this but if it's in front of a sofa, as it commonly is, you won't get a perfect match. Also I've done a repair on a belgian wilton right in front of a french door, and the surrounding carpet had faded a bit. The new tufts, taken from the edge of the carpet, were still the original colour. The customer was still over the moon, even if I wasn't.
In both cases the area will "blend in" over time, and as long as the customer knows what they're going to get and they are happy with the compromise there's no problem.
Partial inserts are a last resort on tufted carpets when a stain can't be removed. The primary backing is cut away, thus removing the tufts as well, then a latex bed is placed down on the secondary backing and a matching piece is cut from an off-cut or under a sideboard or something, then grafted in. I've never done this "live" before, but practised it on scrap carpet a few times. In cases where I could have used this technique, normally the customer has wanted to go down other avenues if they've ended up with an unmoveable stain. It does show up , either a little or a lot depending on the carpet, and would not be too durable in a traffic area. But, for example, if it was a lily pollen spillage next to a fireplace in a part of the carpet that didn't get walked on, it may be viable.
I remember posting a picture a while back, it showed some of my repair kit, got to go out now will have a look see if i can find it in a bit