Many businesses in Rome shut for the entirety of August while people go on vacation – up to 80% of the entire population of Italy is on vacation for the month. Most government offices are shut, as are many independently-run businesses such as small grocery stores, family restaurants, etc.
If you’re a tourist this shouldn’t affect you too badly as most places that cater to visitors remain open, but August 15th is Ferragosto, the culmination of this holiday – it’s a huge family celebration, at least as big as Christmas, so you will find that many things that were open the day before and after are closed on the 15th. Supermarkets are closed, as are many of the restaurants that remain open during other days in August.
We recommend that you shop around beforehand to see which restaurants will be open on that day so you don’t miss out – though obviously hotel restaurants etc. will be open. Here’s a list of restaurants that will definitely be open in August, and some even open on Ferragosto.
Additionally in the Christian calendar this is the Feast of the Assumption so you will find that many churches are closed for services, including the Pantheon at 5pm on the 14th August, 10.30am on the 15th, and it will close at 1.30pm that day. St Peters Basilica will be open but celebrating various masses including the papal Angelus. The Vatican Museums are shut on the 14th and 15th August. The Colosseum, however remains open every day in August.
Pro tip: the Cinecittà World theme park is open all day on Ferragosto, and is delightfully empty that day, meaning you will hardly have to line up for any rides. However you will need to make your own travel arrangements to get there – perhaps renting a car the day before.