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Observable universe means that part of the universe which the speed of light (c) allows us to observe today, i.e., including all space from which a signal (such as EMR) need not exceed c in order to reach us today, given Hubble time, the longest time it has had to travel, a period of around 13.8 billion years. Often, the term universe is used meaning this, though there is no way to know what is beyond the observable universe and it may be more of the same, i.e., additional galaxies, etc., which might logically be considered to be "more of the universe". The term visible universe refers to that part of the observable universe from which we receive EMR, i.e., we can "see" any galaxies, etc. The visible universe extends to recombination, which forms a curtain because before that, the universe was opaque to EMR. The term observable universe also includes the space so far away that recombination blocks our view, as long as it is close enough that there's been sufficient time since the Big Bang for light to traverse the distance. In theory, there are neutrinos that have been free streaming since before recombination that we might someday detect and analyze, and gravitational waves would also pass through and theoretically might be detected in the future. The time from Big Bang to recombination was about 378,000 years, just a small fraction of the universe's age, so the visible universe is only slightly smaller than the observable universe.
Regarding the cited size of the observable universe, there are different approaches to characterizing it so you have to infer what the writer means. An obvious size notion is a sphere with a radius in light-years matching the years since the Big Bang (which is around 13.8 billion light-years), but another distance is often meant: an object that we observe in the early universe has since been (according to observed expansion trends) carried further from us by the Hubble expansion, one determination of the limit to the distance to such objects' current positions being about 45.5 billion light-years (presuming the object still exists: we can't know), so the observable universe is often cited as having a radius of 45.5 billion light years. (We cannot observe the current state of galaxies at that distance, but we can see them at a point in their past or see the earlier portion of the universe that evolved into them.) As time passes, more light from further away reaches us, so we see further, to increasing distances, and increasing this radius. However, the expansion we see indicates a distance beyond which any space is expanding away from us at a rate faster than the speed of light (a distance-growth not forbidden by relativity: the effects of the universe's expansion are not limited in such manner), an apparent limit on the future observable universe: no future person at our position will ever have the opportunity to see an object beyond that distance. At some further distance is one at which no signal has reached our position in the past.
An effect is that such objects on the way to slipping out of sight become redshifted to the point that EMR and GW wavelengths, cosmological time dilation will grow without bound. If we want to see one second dilated to a gigayear, all we have to do is wait until the cosmic microwave background is at a sufficient redshift to see this in front of it.