Half Price Day is the third day of every month. The official neopets shops price their stock at half-price, and you can haggle down further from there. Speed is of the esscence on this day-- even more restockers than usual will flock to the shops to get a better price for the items and will be snapping things up faster than the younger players who tend to hobble into the shops at any ol' time. Kauvara's Magic Shop and the Post Office tend to be the most competitive.
The Hidden Tower does not reduce its prices on half price day but is affected by a similar event on the third Wednesday of every month, and the prices are slashed by 3%. This makes a big difference though when buying those really expensive items, with prices reaching in the hundred thousands or even millions of Neopoints! If you're planning on buying from the HT, it is suggested that you wait until that day to avoid overpaying.