In statistical mechanics, the word entropy is used for $k\ln\Omega$ where $\Omega$ is the number of microstates that are compatible with specified conditions, such as a given total energy and/or a given total volume. If no conditions are specified, then it's the total number of possible microstates. Therefore, whenever we talk about (or read about) entropy in statistical mechanics, it is important to specify what, if any, conditions are being imposed.
Here's an example. Suppose we start with an airtight box full of gas sitting inside a bigger airtight box with no gas. If the smaller box breaks open, so that the gas molecules are free to move into the larger space, then the number of possible microstates of the gas has suddenly increased. In other words, if we define $S$ in terms of the number of microstates compatible with the given constraints, then $S$ has suddenly increased, at least if we pretend that the change in the available volume is the only information we have.
(By the way, in quantum theory, we count mutually orthogonal microstates.)
Now: is the entropy of the universe always increasing?
Although there are many possible microstates, the universe is in only one of them. We have only partial information about which one we're in, so many (maybe infinitely many) microstates are consistent with the partial information we have. We can define $S=k\ln\Omega$ where $\Omega$ is the number of microstates that are consistent with the information we have, and this $S$ may change with time, if it is finite at all. I don't think our understanding of the laws of nature are complete enough yet to quantify this, at least not for the whole universe, as noted in this post.
Some related thoughts can be found here:
Explain the second principle of thermodynamics without the notion of entropy