This question is about cosmology and general relativity. I understand the difference between the universe and the observable universe. What I am not really clear about is what is meant when I read that the universe is infinite.
Basically, I think the idea that the universe is infinite comes from considerations of the large-scale curvature of spacetime. In particular, the FLRW cosmological model predicts a certain critical density of matter and energy which would make spacetime "flat" (in the sense that it would have the Minkowski metric on large scales). If the actual density is greater than that density, then spacetime is "positively curved," which implies that it is also bounded - that is, that there is a certain maximum distance between any two spacetime points. (I don't know the details of how you get from positive curvature to being bounded, but as suggested by a commenter, look into Myers's theorem if you're curious.) However, if the actual density is not greater than that critical density, there is no bound, which means that for any distance $d$, you could find two points in the universe that are at least that far away. I think that's what it means to be infinite.
Overall, the observations done to date, paired with current theoretical models, are inconclusive as to whether the actual density of matter and energy in the universe is greater than or less than (or exactly equal to) the critical density.
Now, if the universe is in fact infinite in this sense, it still could have had a big bang. The FLRW metric includes a scale factor $a(\tau)$ which characterizes the relative scale of the universe at different times. Specifically, the distance between two objects (due only to the change in scale, i.e. ignore all interactions between the objects) at different times $t_1$ and $t_2$ satisfies
$$\frac{d(t_1)}{a(t_1)} = \frac{d(t_2)}{a(t_2)}$$
Right now, it seems that the universe is expanding, so $a(\tau)$ is getting larger. But if you imagine running that expansion in reverse, eventually you'd get back to a "time" where $a(\tau) = 0$, and at that time all objects would be in the same position, no matter whether space was infinite or not. That's what we call the Big Bang.
If the basic question is how we define whether the universe is finite or infinite, then the most straightforward answer is that in a finite universe, there is an upper bound on the proper distance (which is defined as the distance between two points measured by a chain of rulers, each of which is at rest relative to the Hubble flow).
"Does it have infinite mass[...]?" -- GR doesn't have a scalar quantity that plays the role of mass (or mass-energy) and that is conserved in all spacetimes. There is no well-defined way to discuss the total mass of the universe. MTW has a nice discussion of this on p. 457.
"[...]or is it dishomogeneous?" -- I don't understand how this relates to the first part of the sentence. You can have homogeneous or inhmogeneous cosmological solutions.
"How can the universe transition from being finite near the big bang and infinite 14 billion years later? Or would an infinite universe not necessarily have a big bang at all?" -- This was asked again more recently, and a good answer was given: How can something finite become infinite?
The universe would be infinitely large now if it started out infinitely large.
If the universe is infinitely large it can still expand, in the sense that the distances between the galaxies can get larger over time.
But we have no way to know whether it is infinitely large. It could be finite.
I think it is possible that when the concept of "infinity", which is a mathematical concept of considerable subtlety, is applied to physical things such as number of galaxies, it may be that we do not really know what we are talking about. It may be a useful way to make progress by simplifying certain kinds of calculations. This is a common method in physics: we use a potential well with infinitely high walls, for example, or a wave with perfect frequency and therefore infinite extension, and delta functions and things like that. These are all useful as mathematical techniques but we do not need to think there really could be a well of infinite depth, a wave of infinite length, etc. Similarly, in cosmology, infinity can be a helpful way to simplify away various issues which we think are not central to whatever is being studied.
A final remark. You often see it written down, in this context, that if the universe were homogeneous (on average at large scales) and had flat or negative spatial curvature on average, then it follows mathematically that it would be spatially infinite. This is not true. The intrinsic curvature does not dictate the large scale topology. You can have a finite space with either positive, zero, or negative curvature. However, it is fair to say that when the curvature is zero or negative then the finite (or "compact") spatial topology feels less natural, and involves a loss of isotropy at the largest scales.
