single layer vs double layer PCB?
Single sided and double sided is a matter of routing and design convenience (as well as improved circuit implementation quality) at the cost of a more complicated manufacturing process. This difference in complexity has largely dissapeared in the commercial fab world and single layer boards have virtually no price difference with double layer boards. You can have a run of a few reasonable 2 layers boards done with a 2 week lead time, with pretty decent capabilities (8 mil traces 12 mil holes) at a large number of board houses these days for under $100 USD. 

If a hobbyist or student finds themselves making convoluted routes or more than a handful of jumpers to achieve the layout on one layer a two layer design may be worth their time. It gives you flexibility in routing and more breathing room for things like proper ground connections and decoupling and other electrical considerations. However, all of this is predicated on how complex your circuit is, if it fits on layer then stick with that.

I disagree with the other answer in that it  is fairly straightforward for a hobbyist  to  make small runs of double layer boards with a few basic tools and a good amount of care. Design considerations for home making these boards for success is to avoid 90 degree bends and stick to 60mil traces until you can verify your home-manufacturing capabilities. 

tools 


Drill press or dremel tool (ok  for smaller boards) with small bits (1/16" and less) 
Strong UV Light for exposing the photoresist (verify with kit for best hardness for exposure)
Heavy Duty containers for etching 
Decent Laser Printer for printing the stencils.
Hole Punch and Hammer for registration
Dark Box (can be made from cardboard  and thick curtain)  for UV exposure.


The basic process is photolithography using a photoresist  treated PCB  (sold as a  kit usually with chemicals included on the cheap) and a PCB layout stencil printed on transparency with a  good laser  printer.

procedure


Design registration holes into your layout, 1/16" diam pad or so. These pads will be used to align your board on either side with the layout for the photolithography 
Board stenncils are printed on a laser printer with high dpi settings. Multiple sheets stacked  may be necessary, the more sheets the stronger UV light you need.  Make sure to keep track of the mirroring. Make sure that you have enough sheets (or dark enough printer setting) to completely block out the light where your traces are)
Use a sharp hole punch to pierce the printout at the registration holes,  use double sided tape to stick one layout to a side of your board.  Drill through the punched holes and through the board, taking care not to flex the board and use a sturdy clamp to keep the board level (A well cared for milling machine works very well for this process).
use a nail/thumbtack or other dowel to align the other side stencil, afix with double sided tape.
Place the assembled kit into the dark box and verify that the UV light is evenly covering the area where the board will be (can be held on any small stand).
Remove the PCB stencil from one side and peel the protective layer on the photo resist (follow kit instructions for priming as they vary).  Replace the stencil on the dowels and place into the dark box for exposure
After the kit specified time (typically a few minutes), repeat the procedure with the other side
After the correct time, proceed with the development (soaking in one type of chemical) using a heavy duty container.
Following the kit instructions, wash your developer off the board and proceed with etching.
After etching and cleaning, carefully drill your vias with 1/16th holes. Fill vias that are bridge vias (no components) with wire (poor mans through plating) ,   end of a TH resistor works well. Tin the traces with lead free solder clean with flux remover.


The accuracy achievable by this method depends primarily on how dark you can get your photomask stencils. Refraction at the edges causes a smearing that shrinks the traces a bit. The alignment largely depends on the quality of the printing and the drilling. 1/8" vias with 1/16" holes and even a few 20 mil signal traces is achievable with basic office printer. The other limiting factor is the amount of time spent etching, incorrect exposure time or excessively thick stencil will leave some photo resist on etched areas, requiring more time for etching and more likelihood that the etchant will get through the photoresist on your traces. 

The registration is key and is largely a matter of the care with which you mount, drill, and handle the PCB prior to exposure.