Modern commercial fishing uses many different techniques depending on the species sought. Both trawling, nets and line with auto-baited hooks are utilized. Additionally, there are many different types of processing done commercially. Some would require supervision to be certified kosher and some do not (meaning the supervision happens further down the line from the actual fisherman). In cases where the catch is gutted, filleted and flash frozen on the ship, all the inspection requirements mentioned in the OU link in NJM's answer are necessary.
The OU link is primarily aimed at end user (the actual consumer), not the commercial fishing industry. Commercially, inspection usually is happening at the canning/packaging level because that is where the gutting and filleting takes place.
Where baits are used, they would need to be removed which usually happens at the gutting level of processing. So too, non-kosher species need to be removed to comply with the Torah requirement stated in Vayikra 11:9-12. This is why skin with identifying scales attached to each fillet are usually required for filleted, packaged product.
At the consumer level, inspection for parasites does need to be performed, particularly if whole fish are purchased. Most parasites, when present, are found in the digestive tract or the gills. They too are removed during gutting or removal of the heads during processing. If parasites are discovered in the fillet, it should not be used. This would be an indication of infestation of that fish.
On rare occasion, parasites can be found in canned, fully processed fish like tuna and albacore. Since canned fish is actually cooked in the can, something called retort cooking, they shouldn't be used if parasites are found. The size of the parasite would in most cases exceed the 1/60 proportion compared to the tuna in that can and would not be considered nullified.
There are special techniques to handle fresh, kosher fish that have been processed together with non-kosher fish to remove the non-kosher fish oil from the surface of the kosher species.