arduino
First of I have no experience with electronics so I am not sure what I should be searching for to get this done.
All I have with me at the moment is a arduino and a few connectors and LED's. Is it possible to switch the kettle on/off using just these basics (i.e. I dont want to buy more stuff for now).
The kettle is the most basic kind , i.e. the ones that plug into a wall outlet and have a single switch to turn it on or off. Let me know if you need more info.
If you want the thing you're making to be a "proper" Arduino shield, you can buy something like this "Protoshield" board from Adafruit:
http://www.adafruit.com/products/51
Other vendors sell similar boards. It's basically a bit of perfboard that's designed for mounting components, but with the header pins already setup to "stack" onto an Arduino.
If you don't need an actual shield that stacks onto your Arduino (that is, if you're going to run cable of some sort from the Arduino to the rest of the device, or if you're going to use one of the the super small Arduino clones) then you have a couple of options:
Build it on perfboard. You can get perfboard that's designed to map very closely to the experimenters breadboards, or perfboard with one small copper pad per hole.
Make your own PCB (printed circuit board), using one of a couple of techniques. You can use chemical etching, after you either draw traces on the copper clad board with a resist pen, or use the laser printer technique mentioned by Majenko. In either case, it involves soaking the copper clad board in an etching agent that dissolves all of the copper, except the part covered with "resist." What's left are your PCB traces.
Or, if you have an active hackerspace or hobby electronics community in your area, you might get lucky and find somebody who has built a DIY PCB Mill, who could mill the board for you. Or if you're really enthusiastic, you could build your own DIY PCB Mill. :-)
Farm out the "fab" of your circuit board to a specialist. This involves creating the PCB layout on a computer using something like Eagle, and then submitting the file to the fab, along with payment, and then receiving a package of boards a few days later. This may be a little too expensive if you only need one board... this is more for people who are building things to sell in volume. But some hobbyists make it economical by pooling their orders and splitting the cost.
What are those "larger components"? The only larger thing is the relay, and most relays will fit on a breadboard.
This is how you control the relay (the coil is shown next to the diode), it assumes you can connect the 12V's ground to the Arduino's. Resistor, transistor and diode are normal, small components. This relay is just a few cm long, wide and high. It can switch 10A and 230V. If you tell us more about what you want to switch I can give you more directed advice.
edit re your shopping
The relay requires 90mA from your 5V power supply. That will add a couple of hundreds of mW in the Arduino's voltage regulator. At 12V in that would be 630mW, which is a pity. If you have 12V in it would have been better to use that for a 12V relay.
The TIP31 transistor is overkill. It's a power transistor, and they don't have very high \$H_{FE}\$ (the current gain). Next time go for a TO-92 general purpose transistor like the BC547. The BC547B variant has an \$H_{FE}\$ of minimum 200. Go for a high \$H_{FE}\$. This one is still OK at an \$H_{FE}\$ of 100, but I would take a safety factor, and calculate with 40. Then the base current has to be 90mA/40 = 2.25mA. A 1k\$\Omega\$ base resistor will give you 4.3mA, so that's OK.
Best Answer
There's a particular problem here with the kettle that a relay will not solve. Kettles always have their own switch which trips off when they boil. So even if your relay is on, the kettle will not heat up unless this switch is also on. If you took the kettle apart and shorted out the switch, it would not trip off when the boiling point was reached, so this is very dangerous. If you think you can also replace the boiling point sensor and run the logic through your processor, you should think about the fail safe implications of this and consider your house insurance situation.
Depending on the type of switch used in kettle, it might be possible to fill it and switch it on mechanically, waiting for mains to be switched on by the micro. But you could only boil it once before having to mechanically reset the switch. If that works for you, fine, but bear in mind that kettle are not like lamps that you could just switch on and off whenever you like, and interfering with the switch is hazardous.
Most kettles that I have come across do allow the switch to be on even when there is no mains, but most toasters don't allow the basket to be latched in the down position without incoming mains, so you can see this could be a problem.