Usually a real short will trip the breaker. This sounds like an overload plus a defective or over size breaker. Somebody may have solved the breaker tripping by putting in a bigger one. No, no, bad dog! Shut the breaker off and do some investigating. It may well be something you can fix with help from the better people here. It is hard to see how a few lights could over load even a typical 15 amp #14 wire circuit. I did fix an over loaded 20 amp circuit at my church by replacing dozens of 60 watt bulbs with 40 watt ones.
Check the lights on the circuit, make sure there aren't some outside ones too, adding up the wattage. If it is more than 1440, you have found the problem. If not, go to the breaker box. What size is the breaker? If more than 15 amp, try to get access to the cable. It must say on it 12 gauge for a 20 amp breaker, and 10 gauge for a 30 amp one. The switch also has an amperage rating on it to check. The breaker, the wire, switch, etc. all need to be 125% of the load. Watts/120volts = amps. Certainly you don't need an electrician to put smaller light bulbs in. If you need the extra light, look at compact fluorescent bulbs. Even replacing light fixtures is common for DIY.
From there it gets tougher. Regardless of the wire size, the breaker needs to be replaced. A 15 amp breaker that didn't trip before a #14 wire overheated should be trashed. Replacing it is not a good DIY starter project.
If the above doesn't fix the problem, Post back. You may or may not be able to rat out the problem yourself. And by the way, it makes it easier for us, if you keep all the questions on the subject in one thread.
By the way, if the circuit is defective, you may be able to make the seller pay to have an electrician fix it.
To do simple checks like this you do need some tools. A test light, a meter, or a voltage detector might be the best place to start with. I came across the niftiest gadget for trouble shooting, a voltage detector. They work through the insulation of wires. There are several brands. I have a GB Instruments GVD-505A, less than $15 at Home Depot. Touch it to a hot wire, and the end glows red. Find the doodad that lights it on one side, and not the other, and you have the culprit. You do not have to open up housings and expose electrical contacts. You are looking at where your hand is, not where the meter is. Most people are capable of doing repairs and will get it going and not get hurt if they use a little sense. The voltage detector makes it even easier.