Basically which type of board you use depends on what you want to do with it.
Breadboard - About 2cm thick. Comes in a range of colours and has a matrix of holes all over it. It is split into three sections. The top section all holes that are horizontally next to that hole are joined together underneath. The middle section features holes that are joined vertically. The bottom section is the same as the top.
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You can then place your components into the holes and they will be joined together according to which way the underlying copper tracks run. Breadboard is only really suitable for prototyping as it is very large and cumbersome compared to a small piece of PCB.
Strip Board - Also known as veroboard this is a piece of plastic with copper tracks arranged in a lattice. From what I understand you solder your components to this board and then break the tracks with a stanley knife or a scalpel. I really wouldnt use stripboard as it doesnt give you the final finish quality of PCB and it isnt hat great for testing either.
PCB - Printed Circuit Board. This is definatly the way to go I feel. It is very easy and can be done quite cheaply. There are two methods you can use. The first is to draw your design onto the PCB with a UV resistant pen (you will need to use Copper Clad UV light sensitive board). The second, and the much better way, is to design your circuit on the computer and have it convert this to PCB artwork. You can then use a laser printer to print this design off. Spray the piece of paper with WD40 so it becomes transpaprent. You can then place this on your board in a lightbox and you will have perfect tracks for you to etch. If done properly your design will look exactly like it did on the computer screen.
You then need to expose this to light and then place it in acid. Although special etching tanks can be used, you can just mix some chemical together and etch your circuit in a plastic container. You can pick the acid up in ready made bottles, or in crystal form. Both are really cheap and easy to use. Most come with instructions too. I know that Maplins also sells a begginers etching kit for about £25 with some chemical, some board, a pen and an etching container.
As for cutting, Breadboard is not designed to be cut, it is sold as is and would really defeat the object for you to cut it. Several breadbards can also be slotted together to make a larger board. As for stripboard and Copper Clad board, you can cut these just as you would normal plastic. For bests results you should use a gullitene if you have access to one. (No not the type the use to cut peoples heads off, the type of gullitene you would find in a workshop).
So to sum up, if you are serious about making little gadgets then you should invest in a begginers etching kit and some PCB copper clad board. This will set you back about £30. There is also plenty of sites on the internet to guide you through this process. If you just want to mess about and arnt concerned with size then breadboard is probably the way to go for some projects, at least until you get started. Personally I'd avoid stripboard as it really doesnt have any huge strengths over breadboard or PCB.
Hope that helped,
Mike
*** Formerly Code Monkey ***