With most projects, circuit design is a two-part process. Firstly, the actual electronic schematic must be created: this is where you decide which components are going to be used, suitable min and max ratings and so on, and how they are to be connected together. This is the bit where you take a fantastic idea and put it down onto paper.
The second step is to actually lay the components out onto a board (ready for soldering). You can of course stick to solderless breadboards exclusively, but these are usually used for prototyping and testing your schema designs. At some stage, most people like to see their projects develop from the prototype stage to a "completed" stage (you could call this "production stage" if you're planning on making more than one, although a lot of projects at BuildBrighton tend to be one-off ideas).
With an electronics schematic to aid with the layout, you start to place your components onto a board and placing traces to connect each of the components together as necessary. There are lots of applications and tools to help you to do this. One very common app is CadSoft's Eagle. It has lots of very good functionality, including "autotrace" (where your computer calculates a route from each component that needs to be connected, by following your original schematic drawing) and - importantly if you're planning on getting your PCBs manufactured professionally - it can produce Gerber files. A tutorial on how to use the PCB layout side of Eagle is available here: http://www.instructables.com/id/Turn-your-EAGLE-schematic-into-a-PCB
Another very good tool for creating both electrical schematics and PCB layouts is ExpressPCB. The software (at the time of writing) is free to download and use and a fully integrated ordering system is included. The software is generally easier to get started with, and core functions - such as amending library symbols and building your own circuit components - are very intuitive and easy to use and understand. One downside is that because ExpressPCB is given away, to encourage users to order their PCBs from the software suppliers, exporting to Gerber files is not possible. You can, however, print your designs - sending them either to a PDF writer/virtual printer, or to an actual printer. The layout does not need to be reversed before printing, making it ideal for use with Press-n-Peel for home-made, low volume PCB production.
Look out for tutorials on how to use ExpressPCB very soon!