Jaipur has a classic Indian desert-adjacent climate: hot summers, a short monsoon, and a cool, pleasant winter. The best time to visit is October to March, when days are warm (20-28°C), nights are cool, and the sky is reliably clear. January is the most popular month and coincides with the Jaipur Literature Festival and Makar Sankranti (kite festival) — the city's rooftops fill with kite flyers on January 14th, which is one of the more remarkable urban spectacles in India.
April and May are hot (up to 42°C) and increasingly uncomfortable for sightseeing, though the tourist crowds thin out and prices drop. The monsoon runs roughly July through September — rainfall is moderate compared to wetter parts of India, but the heat and humidity make it an uncomfortable time to visit.
Diwali (October-November) is worth timing a visit around: the Pink City is illuminated, and the fireworks over the walled old town from Nahargarh Fort are spectacular. Holi (February-March) is celebrated enthusiastically in Jaipur — if you're comfortable with colored powder being thrown at you, this is one of the better places in India to experience it.