A puncture can happen at any time, it is often caused by driving over something sharp or can even happen from hitting a pothole in the road too hard. A puncture can happen on a brand new tyre and depending on where the puncture is, it may not be able to be repaired.
When looking for a puncture firstly inflate the tyre and then carry out a visual inspection to see if you can find anything sticking in to the tyre. If not then spray some soppy water on to the tyre and rim to see if you can find any escaping air (shown up as bubbles).
It is easier to carry out all of these inspections whilst the wheel is off the car, so if at all possible, then remove it prior to inspecting.
If you do find a puncture it may be able to be repaired but only if it meets certain criteria. If for example he puncture is within one inch of the tyre wall then it should not be repaired. Some garages will do it but it is not thought to be a safe option.