Attraper vs. rattraper
In French, both attraper and rattraper can mean to catch an object in flight, but they are used in different contexts. The choice between the two depends on whether the object was intentionally thrown to you or if you are saving it from falling to the ground.
Attraper
The word attraper is the standard verb used when you intercept and grasp a flying object that was intentionally thrown or launched towards you.
Le joueur a réussi à attraper le ballon d'une seule main.
(The player managed to catch the ball with one hand.)
Essaie d'attraper les clés que je vais te lancer par la fenêtre.
(Try to catch the keys that I am going to throw to you out the window.)
Elle a couru pour attraper le frisbee avant que son chien ne le prenne.
(She ran to catch the frisbee before her dog took it.)
La mariée a jeté son bouquet et sa sœur l'a attrapé.
(The bride threw her bouquet and her sister caught it.)
Il doit attraper la balle de baseball pour éliminer le batteur.
(He must catch the baseball to eliminate the batter.)
Rattraper
The word rattraper is used when you catch an object to prevent it from hitting the ground, or when you recover an object you fumbled or dropped. The prefix in rattraper implies rescuing a falling object or catching something again.
Le verre a glissé de la table mais j'ai pu le rattraper juste à temps.
(The glass slipped from the table but I was able to catch it just in time.)
Le bébé a lâché sa tétine et la mère l'a rattrapée en plein vol.
(The baby let go of his pacifier and the mother caught it in mid-air.)
J'ai fait tomber mon téléphone depuis le balcon mais mon ami l'a rattrapé en bas.
(I dropped my phone from the balcony but my friend caught it at the bottom.)
Le jongleur a failli faire tomber une quille mais il a réussi à la rattraper in extremis.
(The juggler almost dropped a club but he managed to catch it at the last second.)
Il a jonglé avec le savon sous la douche pendant quelques secondes avant de le rattraper.
(He juggled the soap in the shower for a few seconds before catching it.)
Summary
To summarize, you should use attraper for a standard throw-and-catch scenario where an object is passed to you through the air. You should use rattraper when you are catching an object in flight to save it from crashing to the floor, or when recovering something that unexpectedly slipped from your grasp.