Capacity is an informal term. It's very close (if not a synonym) for model complexity. It's a way to talk about how complicated a pattern or relationship a model can express. You could expect a model with higher capacity to be able to model more relationships between more variables than a model with a lower capacity.
Drawing an analogy from the colloquial definition of capacity, you can think of it as the ability of a model to learn from more and more data, until it's been completely "filled up" with information.
There are various ways to formalize capacity and compute a numerical value for it, but importantly these are just some possible "operationalizations" of capacity (in much the same way that, if someone came up with a formula to compute beauty, you would realize that the formula is just one fallible interpretation of beauty).
VC dimension is a mathematically rigorous formulation of capacity. However, there can be a large gap between the VC dimension of a model and the model's actual ability to fit the data. Even though knowing the VC dim gives a bound on the generalization error of the model, this is usually too loose to be useful with neural networks.
Another line of research see here is to use the spectral norm of the weight matrices in a neural network as a measure of capacity. One way to understand this is that the spectral norm bounds the Lipschitz constant of the network.
The most common way to estimate the capacity of a model is to count the number of parameters. The more parameters, the higher the capacity in general. Of course, often a smaller network learns to model more complex data better than a larger network, so this measure is also far from perfect.
Another way to measure capacity might be to train your model with random labels (Neyshabur et. al) -- if your network can correctly remember a bunch of inputs along with random labels, it essentially shows that the model has the ability to remember all those data points individually. The more input/output pairs which can be "learned", the higher the capacity.
Adapting this to an auto-encoder, you might
generate random inputs, train the network to reconstruct them, and then count how many random inputs you can successfully reconstruct with less than $\epsilon$ error.