Chain Tool Selection
Of the three chain repair tools shown, only one of them works. The first tool takes chains apart and puts them together real well but has no provision for adjusting the pin tension after putting a link back together.

The second tool is maybe for older chains.  The problem is that the pin for taking the chain apart, is too large for the modern day bike chains.

The third tool is a marvel and works really well.  Its operation will be shown in detail.

Chain Links
For demonstration, a small chain of links will be used as shown in this photo.  This chain repair procedure works equally well on a chain still on the bike.  

A chain link consists of  two pieces, as shown in this photo.  If you want to lengthen or shorten a chain for instance, this is the smallest allowable increment.  This is important to note as if you are upgrading to a new chain or replacing an old  chain, the new chain has to have exactly the same number of links as the old chain, otherwise the rear derailleur may be stressed beyond its capacity to adjust the chain tension to the shifting task at hand.

For instance, if the chain is too short, the rear derailleur may be broken by the tension of the chain being too tight.  If the chain is too long, then the chain will bang up and down on the wheel support brace even possibly making it impossible to switch gears.

Removing a Chain Link
For this tutorial, the "Park" chain tool is being used.  The "Ritchey" tool could have been used also and would be applied in a similar manner.

This photo shows the link to be removed with its pin underneath the pin of the chain tool press.  It doesn't matter which side of the chain you choose to put into the tool.  The tool is then screwed down against the pin until the tool bottoms out or stops turning.  The tool is designed to stop short of removing the pin entirely for the convenience of reassembly.

This is what the link looks like after pressing the chain link pin out. Notice that the pin is still captured in the link.  To remove the link from the chain, simply apply a little up or down pressure to separate the link from the chain pin and slide it out.

Installing a Chain Link
Now that the link is removed, lets reassemble it which is basically a reversal of the removal process.  If replacing a chain, first thread the chain over the normal route with the protruding pin on the outside of the chain. Then position the assembled link on the chain tool as shown in the photo, with the chain pin pointing towards the chain tool pin.

Now press the chain pin back into the link by screwing the chain tool pin down against the chain pin.  There is no automatic stop when doing this.  You have to look at the chain pin and stop pressing it in when the chain pin on the top is nearly flush with the link.  Now you can remove the link from your chain tool.  

No, you are not done yet.  If you try to flex the link, you will notice that it does not flex but is frozen stiffly in place.  To make the link flexible again, you have to use the chain tool one more time.  Position the frozen chain link back in the tool, upside down from the first position. The object is to press the pin out in the other direction which will unfreeze the link.

As you can probably imagine, this is a very small turn on the chain tool.  You know you are done when the chain link freely rotates while still having both sides of the link still showing a little bit of pin. In another words, easy does it.  In the photo below, the pin on the left is what it looks like before unfreezing the link.  As you can observe, it doesn't take much effort of the tool to push this pin down to a normal almost flush position, as shown by the pin on the right.  (Please observe the note on the above photo.)

Incidentally, chain tool #1 does not have any provision for doing this. Chain tool #2 would work but this chain has a smaller pin size then what tool #2 is designed for.

Now that you know how to remove a single link, you can apply this to a new chain, in order to adjust it's length to the correct length needed for your bike.  As you already probably discovered, your old chain won't come off of your bike without breaking a link. Your new chain has to be assembled together while it is on your bike as well.  This requires a little dexterity as instead of reversing the link in the tool to unfreeze the assembled link, you have to reverse the tool instead.

If you have a tool kit, carry this chain link tool in it as well.  It is rare that a chain would break but if it does, it is impossible to repair a chain on the trail without a chain link tool.  Over the course of about ten years, I maybe used a chain link tool 2 or three times.  Since a broken chain usually destroys a link, I also carry spare links.  It is possible to remove a link and reassemble the chain short one link, but the full range of gears will not be available so you will have to resist using the biggest gear front and back until the missing link is replaced.

Chain Tension, Derailleur
Chain tension is automatic on a bicycle with a rear gear derailleur.  All you are suppose to do and need to do is place the axle as far back as it will go in the axle frame slots and then tighten the axle quick release or axle nuts, whichever you have.  If you need to replace the chain, it is important to note that the new chain has to be the same width and length as the chain it is replacing.  If it is too short and you shift into the largest ring in the front and the largest gear ring in the back, the derailleur could be stressed and may break.  If you are on the trail, and you remove a link to repair a broken chain, make sure you do not use the large ring in the front (high gear) and the large chain ring in the back (lowest gear) or else the derailleur may be stressed beyond breaking limits. 

Chain Tension, No Derailleur
On a bike without a derailleur, you need to pull the wheel into the axle frame slots with the chain properly placed over both the front and rear sprockets.  You would then pull the wheel against the chain until it stops.  If the axle hits the frame stop first then the chain is too long and no adjustment is possible.  Simply go ahead and tighten the wheel down in place and hope that the chain is tight enough not to be thrown off of the sprockets while the bike is being ridden.  Alternately, you can remove one link from the chain and then try again, repeating until the chain fits properly as shown by the axle fitting into the axle slots without hitting the rear of the axle slots.

If the axle does not hit the back of the axle frame slots then back off a little allowing the tiniest bit of chain slack and then tighten the axle down ensuring that the wheel is centered in the frame.  If properly centered, the wheel when spun, will have equal clearance on both sides of the tire and any frame support.  Keep in mind that the axle nuts have to be tight enough to withstand the full pressure of pedaling.  If they are not tight enough, the axle will pull forward when pedaling and the tire will rub against the frame. Also, if you did not allow for enough chain slack, it will make pedaling very difficult and sometimes if extremely tight, impossible to pedal.