Bourg-Madame is a small place in France. It's very close to Spain, so close that you can in fact walk across a bridge in what seems to be the middle of the town and you'll be in Spain. This small mountain town seems to have nothing to offer. But it has, ask the locals for the river with naturally carbonated, pure mountainwater. It's quiete an experience.
For people coming on the N116 from the Perpignan area this is a regular drop-of point since most rides continue to Spain while you go for the fairly less trafficed road to Andorra.
What you wanna do, besides stocking up on food before entering Andorra where the food will be bad and expensive, is follow the signs labelled Andorra out of town for about 10 minutes. You'll come to a rest stop that is a bit too easy to hitchhike from. Easy to cars to see you from a great distance, extremely easy to pull in and snow-cowered mountains in the background all contribute to making this place great.
If you're going to Andorra you'll come to a road junction. N20 goes through a tunnel while the smaller N320 goes over the high mountain pass and down again before going up to Andorra. The latter is recommended.