An application is a program. For the free account, you can have up to five distinct programs installed. If you want to upload a sixth program, you have to remove one of the first five. Each program can have multiple users accessing it simultaneously.
Active time is related to how long a programming is running on a server. Since there is no formal mechanism for "logging out" of a Shiny app, the server automatically "sleeps" a program when it has been inactive for a certain amount of time. The default idle time is 15 minutes, but you can change that for each of your programs. Suppose you have a program that is inactive (nobody using it). Person A goes to its URL, causing a server to spin up an instance, say at 1300 GMT. The program is active as of that time. Suppose that person B starts using it while A is using it, then A stops at 1320 GMT, and then B stops at 1335 GMT. Using the default timeout of 15 minutes, at 1350 GMT the server sleeps the instance, and it counts as 50 minutes of "active time".