A lot of people come to Paris in the summer, which is exactly when a lot of Parisians leave! While the temperature can soar - in fact, we're currently in the middle of a canicule (heatwave) - and air-conditioners are rarer than drinks with more than one cube of ice (and the métro can feel like a sauna...), it's a great time to come to the city in spite of it all. In fact, August is my favorite time of year in Paris.
True, many shops and restaurants close, but you'll have most of the city to yourself. But if you want to be around people, the road next to the Seine is shut down as Paris Plages makes it annual appearance, which makes for a nice place to stroll and relax by the flowing river. And all the outdoor markets are open, although many of the vendors take off for some, or most, of August, too.
My advice is to not worry about getting into the hottest restaurant in town, or what chocolate shops will be open (because of the heat, most close), although internationally known places like La Maison du Chocolat stay open. If you take a walk on the rue de Bac, which has become a haven for pastry and chocolate-lovers, you can see what's open and what's not. The nearby Le Grande Épicerie for sure will be, and it's got AC!
And even though the famed Berthillon closes in August (the owner once famously said something along the lines of - "We don't want people eating our ice cream just because they want to cool down"), a lot of other places around the Île-Saint-Louis serve it, as do some cafés in Paris. But there's most to the ice cream scene in Paris than Berthillon. Check out the list on my site of Paris ice cream shops.
Here are a few links to help you find things to do this summer in Paris:
Summer Festivals in Paris. (Secrets of Paris)
List of Outdoor Paris Markets (Paris.fr)
What's Open in August: List from 2014 but likely to be updated and probably the same restaurants will be open in 2015. (Paris by Mouth)
Top Events to Look Out for in Paris 2015 (Paris Visitors Bureau)