Before the answer, the thing that actually matters: whether the keys are weighted matters far more than how many there are.
A weighted key answers differently depending on how you play it — gently for a quiet sound, with the arm's weight for a full one. That relationship is what playing the piano is, and everything I teach about tone and evenness depends on it. An unweighted keyboard, where every key gives the same volume however you play it, teaches a hand to poke, and poking then has to be un-learned on a real instrument.
So: sixty-one weighted keys beat eighty-eight unweighted ones, every time and without argument. Now, the counts.
61 keys — five octaves
What it does: the whole of a beginner's first year, comfortably. Method-book pieces live in the middle of the keyboard, and 61 keys give you generous room on both sides of that.
When it runs out: sooner than people expect, and not gradually. It runs out the day you open a piece of real repertoire, because classical writing uses the extremes deliberately — a bass note two octaves below the hand, a melody carried up into the top. When notes are missing, they cannot be fudged; the piece simply cannot be played as written.
Buy it if you are testing whether the habit will take, you have very little space, or you need to carry it. Do not buy it as your one instrument if you intend to keep playing.
76 keys
What it does: a great deal, and for longer than 61. Most early and intermediate repertoire fits.
The honest problem: it is an awkward size. It costs close to what a full 88 costs, takes nearly the space, and still runs out eventually. I rarely recommend it, not because it is bad but because the money is better spent on the full set.
88 keys — the full piano
What it does: everything. Every piece written for the instrument, forever, with no adjusting.
There is a second reason beyond repertoire, and it is the one people do not anticipate. On a full keyboard, middle C is where middle C is. Your hands learn the geography of the instrument — where the notes sit relative to your body — and on a shortened keyboard that geography is wrong. Move to a real piano and everything is displaced. It is not fatal, but it is a needless piece of re-learning.
Buy it if you are buying once and keeping it. Which, if you are reading this far, you probably are.
So, plainly
If you already own a keyboard of any size: start today on it, and do not think about this question for six months. Beginning now matters more than any of this. The only thing worth checking immediately is whether the keys respond to how hard you play — and if they do not, plan to replace it before your second year rather than your second month.
If you are buying and can only buy once: 88 weighted keys with hammer action. Nothing else on the specification sheet is worth the money: the six hundred voices, the accompaniment styles, the light-up keys and the built-in lessons are all ways of not practicing. One good piano sound and a key that answers your hand.
If the budget will not stretch to 88 weighted: buy 61 weighted rather than 88 unweighted, and plan to upgrade. You will have learned the right physical habit, and habit is much more expensive to change than hardware.
The purchase nobody puts on the list
An adjustable bench, and a stand that does not wobble.
I mean this seriously: a bench will do more for how you sound than the difference between two instruments at the same price. Almost every fault I see in adult beginners begins with sitting too low, because a dining chair is built for a table and a table is lower than a keyboard. And a keyboard on a flexing X-stand moves when you play with weight — so the hand quietly learns not to use weight, which is the one thing you most need it to learn.
Solid stand, bench that adjusts, weighted keys. In that order of neglect, and roughly that order of importance. The rest of the specification sheet can be ignored.
My notes on instruments and the book I teach from are on my books and gear page. But do not let any of it delay you: the best instrument for an adult beginner is the one that is in the room, today.