Download speed is how fast you can pull data from the server to you. Most connections are designed to download much faster than they upload as the majority of online activity, like loading web pages or streaming videos, consists of downloads. Download speed is measured in megabits per second (Mbps).
Upload speed is how fast you send data from you to others. Uploading is necessary for sending big files via email, or in using video-chat to talk to someone else online (since you have to send your video feed to them). Upload speed is measured in megabits per second (Mbps).
Ping is the reaction time for your connection - how fast you get a response after you have sent out a request. A faster ping time means a more responsive connection, especially important for applications where timing is everything (like video games). Ping is measured in milliseconds (ms).