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 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., 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 (though not opaque to gravitational waves, which, in theory we can still detect). 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. 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 (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. As time passes, more light from further away reaches us, so we see further, to increasing distances, and increasing this radius. However, the current expansion-trends indicate a distance beyond which any space would expand 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) and there exists a distance beyond which signal has never reached us and will never reach us given the current trends in the universe's expansion. This is 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. Another effect is that signal from the most distant objects are redshifted to the point that any EMR received will be at an extremely long wavelength, growing more extreme with time (and analogously, any gravitational waves from there will shift to much lower frequencies). I've read that viewed objects slipping out of view would apparently freeze in place (given current trends). Arriving photons will be more spaced out, making the objects dimmer, and will redshift without limit. The redshift calculation also quantifies cosmological time dilation: when we view events one second apart at the source, we see them with a longer time between, and with the expansion, this grows without bound: for example, if we want to see one second dilated to a gigayear (or any finite interval), all we have to do is wait long enough, i.e., the ratio time-at-the-source to our time falls toward a limit of zero.