Upload/download speeds tell you how much data will transfer in each direction in a certain period of time.
Ping tells you how long it takes for a packet of data to reach a destination. Obviously, the more network hops a packet has to take, the longer the ping times will be. This is also why you want to run a speed test against a server that is close by, i.e a low hop count (or ping time.)
If you're just browsing the net, downloading email, or that type of thing, ping times really aren't that important.
Low ping times, aka "latency," are far more important if you're gaming online (reporting your activity to the server, and getting other player's info back quickly) or using a Virtual Private Network. VPNs are quite sensitive to round trip times. I can't use a VPN connection to work from my sister's house, she's got a microwave hop to her ISP with a high latency issue.