Removing a hook embedded over the barb, but not back out through the skin, is relatively easy:

First

Make a loop about 10cm (6") long, in strong line, 10kg (20lb) plus, and pass it over the eye of the hook, and then up to the top of the bend of the hook.

Then

Push firmly down on the eye of the hook so the eye of the hook touches the skin.

Finally

With a sudden, strong yank on the line loop, pull up and away from the hook-eye. The hook should come out the way it went in, and because it will curve out the same way it went in the barb should not catch.

It is very important to follow 'Step 1' to the letter if pain and discomfort is to be avoided. Most importantly the loop of line must be at the top of the hook bend, and the pull must be up and away.
Way too many articles on this method advise having the line at the back of the bend, and pulling straight back - this will only result in the barb catching and causing pain.

You can make this hook removal a one-person operation where the hook is in your arm or hand, or anywhere you cannot use your two hands. Make the loop larger and hook the loop around something immovable, like a tree branch, then press down on the eye of the hook as above, and jerk your hand away in the direction shown above. I used this method once when I stuck a hook in the back of my hand, and I hooked the loop around the tow bar ball of my car - worked a treat - felt nothing!

The sooner you do it after the hook goes in the less time there is for shock to set in around the wound and make removal with anesthetic the only viable proposition. If you have some ice handy, pack it around the hook for a minute or two before pulling it out – I actually don’t think it does nothing to reduce pain, because there usually is none, but it sure seems to make the ‘patient’ feel better before you do the deed.

In general it is a pretty painless business; the thought is more painful than the extraction itself. In fact of the many hooks I have removed from myself or others, there is one common factor - the almost complete, and surprising, lack of any pain.

If you are using small hooks, smaller than 4/0, (hook sizes) you can use a pair of long-nose pliers, or forceps. Grab the hook at the top of the bend, and then quickly and firmly rotate your wrist, and the pliers, towards the eye of the hook. This method can be used if you are alone, by using the thumb of your forceps hand to push down on the eye of the hook.

If the hook has gone in and then out through skin, and you have the necessary tools, you can cut off, or crush down the barb, and pull the hook back out the way it went in. Any delay will only make the operation more painful than it needs to be. But this is somewhat painful, and really only an option for small hooks.

Some have recommended pushing a deeply embedded hook right through and out of the skin again, then cutting off the hook barb. In general this is very bad advice. The further pain will make the area go into shock, you run the risk of puncturing blood vessels or nerves, it is extremely painful, and it is an all-round silly thing to do. Big hooks, deeply embedded, require trained medical care.

If you need a good reason to use barbless hooks, the first time you have to remove a hook from yourself or fishing partner, your resistance to this change will disappear.

If you use treble or double hooks, (and I really urge you not to; they do too much damage to fish and fishermen), the first time you get a couple of barbs from a multiple hook stuck in a portion of your person will teach you all you need to know about why not using them is such a good idea.

As you sit in the doctors rooms watching him or her slashing and hacking to get the damn thing out, and then charging you $200 for after-hours attention, say “OK Bish you win, I will never use trebles or doubles again.”

All the above is an excerpt from my book
"Fishing Even Smarter"

A few years ago I stuck a fly-hook into a friend’s throat (don’t ask). I said I would remove it using the forceps method, but he thought he needed to see a doctor. So hook still in his throat we drove the hour back to my house, and he and his wife set off for the hour’s drive to the after-hours on-duty doctor.

They waited another hour and a half to see the doctor, and yes, you have guessed what happened.

I asked him how the doctor had removed it, and he sheepishly told me he had used the quick flick with a pair of forceps method, dabbed on some antiseptic, extracted a heap of dollars out of his wallet even faster than he extracted the hook, and sent him on his way. Total time in the surgery, 2 or 3 minutes!

It nearly choked me, but I managed to suppress the urge to yell "told you so", so he is still a friend.

Just two weeks ago I stuck a fly into the back of my head. I was an hour's walk up a river and on my own. Two options; leave the fly where it was till I got back to civilisation and help, or, try to remove it myself. I chose the second option, as the fly was right on the collar-line and would have irritated me all day.

Feeling behind my head I found the eye of the hook and snipped off the leader. I made a loop in some 8lb leader, pulled the loop under the eye of the hook and up to the top of the hook bend. With one finger of one hand pressing down on the eye of the hook, I yanked on the loop. Then nothing!? No pain. A loop of line in one hand, nothing under my finger of my other hand, and no fly, in my neck or in the loop. Never did find the fly.