moderna and pfizer are mRNA-platformed vaccines. they inject an RNA 'messenger' that 'infects' one of your healthy cell's RNA (not DNA) to produce the spike protein without the actual virus. it's similar to how an actual virus works, but without the actual virus... or something like that
j&j is something else. probably injecting deactivated coronaviruses or something but i dunno. i've heard it's single-shot but only off-hand
it's not known how long the immunization lasts (months to a few years?) or whether it truly prevents infection or just reduces the effects to the point of non-severity
something rather annoying and probably due to quick development is how it really has unique effects to each person... like a game of chance
the difference is that usual vaccines use a vector disease like the common cold that has been modified to include DNA that the cells transcribe into mRNA then it creates the "docking point" of a virus like covid so that your immune system can bind to it.
some vaccines use live viruses where, as you said, its a weakened covid virus that is injected but they remove the part that rapidly duplicates itself so it can spread.
The downside of all these methods is you have to grow and tend to viruses in labs then edit them, this can be extremely dangerous and its pretty hard to mass produce. For example, some vaccines use literally 1 chicken egg per dose to grow the viruses. They then centrifuge the egg parts out but some particles make it through sometimes. This is why a lot of vaccines have exemptions for people who have egg allergies.
mRNA is cool as stuff because it skips all these weird ass ways to make vaccines, all you have to do is synthesize a bit of mRNA which is easy as stuff to do and scale. It also skips the DNA transcription and host virus so the efficacity is boosted through the roof. It's also incredibly easy to modify and adapt for future strains because all you have to do is forget with the mRNA until it matches the mutated spike proteins and you can technically grow anything inside someone using mRNA instructions.
not to mention it completely avoids any kind of vaccine shedding or any chance that a live virus wouldn't be deactivated properly.
it truly is a marvel of science and an incredibly effective way of giving people high immunity (even if we notice that the immunity lasts only for like a year, the fact that covid wont be able to spread as fast will make it die out pretty fast)