Nouakchott, Nouâdhibou and Atâr have international airports. Nouakchott's airport handles most traffic.
The only direct flights from Europe are through Paris, with Air France.
Mauritania Airlines flies twice a week between Nouakchott and Dakar, six times to Bamako (three of which continue to Abidjan, Cotonou and Brazzaville). Senegal Airlines flies five times a week to Dakar. For other Saharan or sub-Saharan countries, you'll have to change in Dakar or Abidjan.
Mauritania is well connected to North Africa. Royal Air Maroc operates between Casablanca and Nouakchott five times a week, while Tunis Air connects Tunis with Nouakchott (three times a week). Air Algérie flies to Algiers twice a week. Mauritania Airlines has four flights a week to Casablanca from Nouakchott, and twice weekly from Nouâdhibou.