Monsoon would be on the full swing in last week of July. Amount of rain will differ depending upon which part of the country you are in.India is a huge country. Rajasthan for example in some parts wont see any rain atall and may be a few drops if they get lucky this year. Nothing stops in India and as long as you stay away from the flood prone areas of eastern UP and Bihar you will be fine. Rains also mean that temperatures will come down. Kerala gets plenty of rain in July and it remains hot and humid throughout but again nothing really stops.Traveling in the hills gets a bit tough and some land slides are not uncommon so if you are planning to go to places like Manali or Leh by road then it is best avoided or just take a flight to Leh and skip Manali.
As you can imagine India gets monsoon every year so the infrastructure is built keeping that in mind and the disruption to traffic is minor. There may be slight delays because while it rains the traffic slows down a little to avoid slipping and may be a few traffic bottlenecks here and there so keep a 10-20% extra margin for traveling time.
I would also like to add that during monsoon it does not rain non stop. Rains may go on for 6-7 hours and then no rain for two days to give you an example.
Source : I live in India (New Delhi)