Here I'll detail what I did to build my low-battery circuit, hopefully some of this will also apply to more general circuit making. First, start with a schematic diagram; here is mine.
This circuit will show a green led when the battery is ok, and switch it to show a red led when the battery voltage is low. There's more explanation at the end of the post in this thread.
The first step is to find out what parts you need, all circuit diagrams should use the same symbols for components. This page http://www.kpsec.freeuk.com/symbol.htm
has a good list of the most common symbols. In my case I need: 6 resistors, 5 npn transistors, 2 leds and a potentiometer. Now that you know what components you'll require, you need something to build the circuit on.
There are a few types of prototyping board available, the type I use is called "Veroboard" or stripboard. It's the type that has rows of copper strips, there are other types that have separated copper pads and are just as useful but quite expensive where I live. I find 9 x 25 stripboard is usually enough to build most small circuits on.
Next you need to translate the circuit diagram to your stripboard layout. This is the tricky part. I usually do this by hand on graph paper, because I find it easier to work with. This page http://www.zen22142.zen.co.uk/Prac/vero/vero.html
has some useful templates that you can use if you want to do it by copy and pasting in paint.
You need to think carefully about where you place each component and where you may need to cut tracks or add wire links to complete the circuit. You have to take into account the size of each part and where the leads are. There is software to do this "automatically", but I find for small circuits it's not really worth the effort in setting up and it always requires manual tweaking anyway.
Here is what I came up with for the above schematic:
The plus and minus symbols are where the battery is connected, the blue lines are wires, the brown squares are where the track underneath is cut and the part on the left is the potentiometer used to set the voltage the low-battery led comes on at. It's important to note the order of the pins on your transistor because they may not be the same as on my diagram, eg. from left-to-right; 2N4401 is E-B-C, and BC547 is C-B-E.
Now to start building the circuit. When using stripboard you should begin by cutting the tracks, this can be done with a dremel or by carefully twisting a screwdriver or drill-bit inside the holes until the copper track is broken. It's a good idea to clean the board after this step to remove any residual copper.
For the low-battery circuit, make the cuts according to this picture.
Next gather the components:
To make it easier to solder, start with the flattest components first working your way up to the tallest. I start by making the wire bridges. I keep the offcuts of components legs for this purpose and bend them into shape with needle-nosed pliers. Alternatively you can use any type of wire for this.
With the copper side down, place the wire bridges according to this pic.
When soldering it will sometimes help to use a crocodile clip to keep the piece in place:
After soldering the wire bridges, trim the leads as close to the solder joint as possible without actually cutting into the solder. If you cut too close it will feel soft, you should hear a loud *snip* as you cut the excess leads(unless you are using plain wire for this part).
When you have done all the wire bridges, move on to the resistors:
After that you can solder the rest of the components.
And there you have it, one finished low-battery indicator circuit.
edit: underside view