Does science have an explanation of how the first "common ancestor" formed in evolution?
Darwin's theory of evolution (and the modern theory of evolution) explains how the current diversity of life developed. Remember, the title of Darwin's book is On The Origin of Species, not the origin of life.

As for how the current diversity of life developed, or how species originate, the theory of evolution states that life forms change and diversify gradually, and species arise and differentiate from related species, via the processes of random mutation at the time of reproduction and natural selection. The ToE is a complex set of theories with many different semi-independent aspects but that's what it basically comes down to.

Clearly all this requires life, or imperfect reproduction at least, to exist; it says nothing about the first life, and it cannot say anything about it because given what we know of the processes that lead species to arise and life to diversify and aquire its form, those processes cannot possibly have been in play when going from abiotic stuff to the first living systems.

In other words, the Theory of Evolution is how life develops once it starts existing; it would work equally well if the first life appeared via natural chemical processes or was created by alien tinkerers or the result of a time paradoxical sandwich dropped in the wrong place on primordial Earth. The study of how life could start existing is usually called "abiogenesis".

The issue isn't actually as clear-cut as it may seem, since there is a very wide unknown space between what we consider the most archaic forms of life, and any entity that could plausibly arise via purely abiotic processes; every theory of abiogenesis does assume that a lot of the features we consider essential to life must have arisen after some kind of replication appeared, meaning those features would have evolved. So there definitely is some evolutionary biology involved in investigating abiogenesis, and maybe if we ever solve abiogenesis it will be folded into the ToE (like I said, the ToE is actually a complex set of theories and observations, not one single thing. So while our understanding of what the theory says and can say currently excludes abiogenesis, our understanding and definition of the theory could evolve). But we haven't, and it currently isn't.

You need to edit your question however, because it is completely unclear from the title or text that you are asking about abiogenesis. Your question sounds like it's about embryonic development or biochemistry. Those are the current instances we have of organisms forming; whatever processes were at work in creating the very first life, well for one thing maybe we wouldn't want to call whatever that was an "organism", but more to the point those processes cannot happen today. The atmosphere is wrong and too full of oxygen, there are organisms everywhere vaccuuming up whatever resources those original biochemical processes might have used, basically there is likely no chemical environment on modern Earth that's anything like the chemical environment life originated in.

To answer your question though, abiogenesis is currently an unsolved question, so no, Science does not have an explanation of how the first organisms formed. But if you want to have an idea of how things could have happened, what the challenges are in figuring things out, and what things Science currently considers likely or impossible, there is a lot of active research in the field and many different hypotheses. The Wikipedia page for Abiogenesis has a fairly comprehensive rundown on this.

This video describes one of them (my favorite and the first I've found actually convincing, I have no expertise whatsoever to base this on but I plug it anyway; if nothing else it gives an appreciation for what kind of things the researchers in this field look at when thinking abiogenesis) :

https://www.youtube.com/watch?v=av98Brx23_4

(disclaimer: youtube throws up an error for this video for me right now but I've watched it before, I don't know if the video's down or if the problem's on my end)