Axis is what connect wheels to a car or other moving parts such as a door.
The connected object will be free to rotate on one axis.
You can make the object spin freely by increasing friction.
If the number is low, but higher than 0 the axis will break pretty easily.
If the number is high the axis can still be broken, but it needs more power to break it.
Set this to 0 if you don't want it to break.

If you know what an elastic is IRL, you know what it is in Garry's mod. The purpose is still the same.
So lets go through the settings.

Elasticity
The strength of the elastic. The higher this number is, the tougher the elastic will be.
So if the number is 0 you can basically use it as a rope.
If the number is 4000 (max) the elastic will use all its "power" to move closer to the other end of the elastic. Just as you know from Real Life.
So try to find a number somewhere between 0 and 4000 for your build.

Damping
The higher this number is, the less bounce your elastic will have.
So, If you move one of your props connected to the elastic, away from the other end/prop using your physics gun, and then letting it go. The prop will attempt to come back pretty aggressive (if the number is low)
If the number is high the prop will "slowly" move back to the other prop

Relative Damping
Pretty much the same as Damping. I'm sure there's a good reason for there being two options (I have no idea what the difference is)

Stretch Only
If this is enabled the elastic will only pull the other prop closer to it.
If this isn't enabled the elastic will try to get the other prop to that point where you created the elastic. It will both push and pull.

Rope Width
Basically makes the elastic thick or thin. The number is 0 you won't be able to see the elastic. (This is visual only)

The hydraulic tool is very useful making a gate, elevator or similar.
By right clicking on a prop, the tool will automatically create a hydraulic with a 90 degree angle from the point you pressed. 
(Look at the picture)       (This will work with any other rope based tools)
[Image: yqzVnAi.jpg?1]

Important
-Do not use Smart-Weld on the hydraulic or the props it is attached or No-Collided with.
-The moving prop may not be welded to a frozen prop. If so, it will no longer move.

Distance
The distance you want your hydraulic to expand by when you press the toggle key
If the number is negative, the prop will move closer to the other prop.
If the number is positive, the prop will more away from the other prop
                                                                                                   
Speed
This is the speed you want the hydraulic to operate at.
If the number is low the hydraulic/prop will move slowly.
If the number is high the hydraulic/prop will move fast

Fixed
(Mostly used) If this is enabled, the hydraulic will move in a straight line.
If this is not enabled, the hydraulic will move freely and not in a straight line.

Width
Basically makes the hydraulic thick or thin. The number is 0 you won't be able to see the hydraulic. 
This option is visual only.

Video

Spoiler :

This tool is almost the same as the axis tool. The difference is that you can make the object you attached to spin forward or backwards.
The motor tool is very useful for objects that needs to spin, such as a wheel on a car or in a casino.

Torque
How fast you want the prop to spin

Force limit
How much force it takes to break the motor.
If you don't want the motor to break set this to 0

Hinge Friction
The more friction the more freely the prop will move.
If you want the spinning object to stop spinning faster after you have released the motor, try to higher this a bit.
This is 0 the most of the times.

Force Time
For how many seconds do you want the motor to operate?
Set this to 0 if you want it to operate forever, or till you press the key/button/keypad to stop it.

The Precision tool is a VERY useful tool.
I use it all the time, Place props, No-Collide props etc.
(If you are missing some of the options below, make sure that your “General tool options” is an Experimental. You can find this at the bottom.

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Apply
[ ] Freeze target: Freezes the Object/Prop. This should be on unless you know what you are doing.

[ ] Auto-Align to world (nearest 45 degrees): Simply Aligns the target/prop to the nearest 45 degrees. Very useful for walls, roof and floors etc.

[ ] Disable target shadow: Simply disables the target/prop shadow. Have in mind that this is only for the prop shadow and not world props. Also, It will not remove shadows on the prop, It will only remove the shadows the prop are creating.

[ ] Only collide with players: This one is very useful, especially for larger props. It will no-collide the prop with the world and cars, but not the players.

[ ] Disable Physics on object: This will make you able to use your physics gun on the prop, but it won’t move the prop. This can be very useful if you have a few people that got access to your props

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Rotate
Rotates the tool.
Use it by left clicking somewhere on the object, depending on where you want to rotate it from -> Now move your mouse and it will rotate. When you are done rotating it, left click again.

*Rotation Snap (Degrees): Makes you able to rotate the target/prop in different directions.

[ ] Freeze Target: Freezes the Object/Prop. This should be on unless you know what you are doing.

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Move
This tool is used a LOT, This is for sure the tool I use the most.
It makes you able to precision move a prop with two simple clicks.

[ ] Rotate Target: Makes you able to rotate the target once you have moved it with the tool.

*Snap Distance: How far away the prop will spawn from the target. Set this to 0 if you want it to be normal.

*Rotation Snap (Degrees): Works like the Rotate Target (Rotate Target needs to be ON in order for this to work)
When you move your prop from A to B using the tool, you will then be able to rotate it using your mouse, When you are done rotating, simply left click somewhere.

[ ] Freeze Target: Freezes the Object/Prop. This should be ON unless you know what you are doing.

[ ] No-Collide targets: If you move your prop onto another prop, the two props will be no-collided (If on)

[ ]  Auto-Align to world (nearest 45 degrees): Simply Aligns the target/prop to the nearest 45 degrees. Very useful for walls, roof and floors etc.

Weld
Everything is the same as the above, The only different is that it welds the two props together

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Axis
Works exactly like the other Axis tool.

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Ballsocket
Can be very useful in some cases, mostly for trailers or similar.
It simply creates a ballsocket between the two targets.
The two props can move freely in any direction, even though they are kind of tied together.
The props can’t move away or closer to each other. It is like the axis tool, it just makes it able to rotate however it wants.

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Slider
Creates a slider between the two targets. Makes them able to move away or closer to each other. They can’t move sideways, up or down or spin. Only the way the slider is pointing.

Weld - smart : Efficiently welds selected props together and/or to the world. NOTE: All selected props will be frozen and have all of their constraints removed (weld/hydraulics/elastics and so on). Auto-select inside radius. The auto selecting inside radius, anything before this value won't be selected, if you want to select everything use '0' for this value  Auto-select outside radius. Anything beyond this value won't be selected. Weld forcelimit: The strength of the welds created. use '0' for unbreakable welds.

If you are having trouble on how to get this to work, Set your settings as follows:
Auto-select inside radius = 0
Auto-Select Outside Radius = 999999 (something really, really high)
Force limit = 0
Weld mode = Props