That topic is so broad but I'll try to summarize. Basically, the lower the ping time, the better your connection. But that doesn't always hold true. The considerations are: is the connection better in terms of response time or sheer throughput (the total amount of data it can transfer)? This brings me to your first question.
I'll start with an analogy for anyone who is seeing the word 'ping' for the first time. Let's say you want to test how quickly your postal company can send a letter from you to another person and back. What you do is to write a letter with instructions for the recipient to send it back as soon as it arrives. When the person receives the letter, he immediately gives it back to the postman to send it back ASAP. You are only interested in the total time it'll take from the moment it leaves your hand to the moment you get it back. Now, that letter in our ping case is called a packet (it's essentially a tiny amount of data of a predetermined size.) Ping time is the time it takes for the packet to move from your computer to the computer (server) you are testing and back to you. It's measured in milliseconds. A millisecond is one one-thousandth of a second. It has to be measured with these tiny units because data travels very quickly.
However, ping doesn't tell the whole story. It only tells you that if, for example, the ping time from your computer to the server you are testing is 20 milliseconds, it will take that amount of time for your computer to send a request to the server and get a response. It does NOT measure how much data the other computer (server) can send you or you to them.
Lower ping time means that you have to wait for a shorter period to receive responses from a server and vice-versa. On the question of which one is better, it depends on what you are doing on your computer. If you're downloading large amounts of data, the ping times are less important than the throughput. In that case, you would be just fine with high ping times even in excess of 500 ms. Bandwidth is more important in this situation because you want as much data transmitted as possible. It makes little difference if it takes 500 or more milliseconds for the data to start 'flowing.'
There are some applications that need low ping times, most notably, games. If you are using online multi player in a game, the game server wants a response from you almost immediately and your game also wants data from the server immediately. In this case, your connection must be a low-latency (with very low ping times) one. Games usually work better with ping times below 50 milliseconds or less. Some can't work unless the ping times are below 100 ms.
In the speed test, ping times show you the server's response times but the speed (throughput) is a totally different tale. I have seen ping times as low as 2 milliseconds when safaricom launched their 4G but now it's much higher although it makes little difference. Fiber optic connections can have ping times lower than one-tenth of a millisecond.