Method 1
This is an update to the existing page. The old method that was found here and on similar sites just wasn't/aren't well thought out. After having a little alternator crisis with my Roadster which resulted in having to remove and refit the alternator 3 times, I worked out a better way.
Jack the car up, remove the rear wheels, disconnect the battery and remove the arch liners on
both sides. Remove the 3 plastic nuts with a 10mm hexagon socket and pull the arch liner out.

Protecting the belts is a plastic cover. Pull out the front plug...

...followed by the rear plug. The cover will now come off the car.

Disconnect the X-frame by removing the arrowed bolts on both sides.
This gives access to the engine mount bolts.

Use a trolley jack and a piece of wood to support the engine under the gearbox.

Take a 15mm spanner and loosen the adjuster nut. Look for the tab circled yellow,
push it in to release the rear cover. Remove the plastic cover from the car.

Pivot the alternator backwards to loosen the belt.

Remove the 2 Torx bolts circled below (E10 and E12). Take the bracket off the car.

The belt is now loose enough to get a 16mm socket onto the main top pivot bolt.
A square nut will fall off the back but you'll not lose it. Slide the bolt out.
It will easily slide out above the engine cradle before you lower the engine.
If you have a magnet on a stick you can position it on the rear nut so it doesn't fall.

Then remove the 2 Torx bolts from the right side engine mount.
Slowly let the jack down that you put under the gearbox, the engine will now tip.

You now have the space to reach into the top of the engine bay and wiggle the alternator off of its pivot point. The grip is quite tight so keep moving and pushing.

Rotate the alternator like this...

...so you can see the connections when you look into the top of the engine bay.
It is very important to NOTE WHAT CONNECTIONS YOU HAVE.
You will have a large wire going to connection B+, this goes to the battery and starter motor.
Nothing will be connected to the central D+ connection.
On the PHIN connection you may have 0, 1 or 2 wires, when you fit a new alternator
make sure that all the wires that were connected are reconnected.
0 wire to PH-IN = self exciting alternator.
1 wire to PH-IN = externally excited alternator (from ECU or ZEE)
2 wires to PH-IN = externally excited alternator (from ECU or ZEE) and diode lead to engine earth.
Disconnect the wires, this is where you are reminded about disconnecting the battery.
If you didn't, if your ratchet touches the earth as you remove the live wire. POP!

You can now angle the alternator as shown and slide it out between the cradle and the suspension.
Other guides lower the engine cradle which doesn't give you as much room to work, so ignore them.

Fit a new alternator in the reverse of removal. Slide the alternator into the engine bay,
reconnect all the wires correctly and offer it up to the pivot points. Getting it back
between the pivot points is as hard as getting it out but again, with a lot of wiggling
and pulling from the top of the engine bay, it'll eventually slide in place.
The alternator pivot bolt should be torqued to 25Nm.
Re-tensioning The Belt
Place a 15mm spanner onto the nut and an E10 socket onto a torque wrench and onto the end
of the bolt. Set the torque wrench to "undo" and dial in 45Nm. Push the handle of the torque
wrench and the alternator will pivot up and the belt will tighten. When the torque wrench
reaches 45Nm you can tighten the adjuster nut with the 15mm spanner.

Turn the engine over and check the alternator output.
Method 2
Thanks to Richard D for this information
Disconnect the battery and secure the car up on ramps, drain the oil and remove the oil sump.
Remove the belt guards and belts and undo the three bolts. Prise alternator from
cradle and lower it enough to disconnect cables. Jack the body up a few inches so
there is enough clearance between crankcase and body to drop the alternator out.
This isn't the quickest way to get the job done but if you need to change the oil, bonus.
Why Are You Removing The Alternator?
If your alternator spins freely but doesn't create a charge, check the diode lead.
This isn't found on all versions of the smart but a good place to start as they do fail.
Seized Alternator
If your alternator is seized you may not need to remove it from the car to fix it.
The open design is to help cooling but unfortunately also allows water to get in.
This can seize the alternator of a car left standing for more than a few days.
Depending on what tools you have at your disposal, you may be able to repair it without
lowering the engine at all. Just place a 24mm socket on the main nut and use a long handled
ratchet to slowly force the alternator to turn. Do this slowly in both directions, 1 to 2 turns
in each direction. A penetrating lubricant can help when sprayed into the alternator case.
When the alternator spins with less resistance, remove the tools and try to start the car.
How Do I Know My Alternator Is Seized?
A seized alternator and a seized starter motor have the same diagnostics. The engine won't turn.
To work out if it is the alternator or the starter motor, try push starting the car.
If you can push start it, it's the starter at fault, if it won't start, the alternator is probably seized.
By printing this document I state I have read and understood this
disclaimer