One of the central factors that defines whether something is alive is its ability to reproduce - to make a new version of itself. The first things we know of that could do that were molecules of RNA, more than four billion years ago. RNA could reproduce, but often the copies were messed up and didn't work right, because some mistake happened during the copying. After a while some RNA molecules developed into DNA molecules, which looked like a ladder, so they had two identical sides that could check each others' work and make sure that the copy didn't have mistakes in it. This didn't always work, but it worked often enough to be a big advantage in reproduction.
By about 3.5 billion years ago, some DNA molecules were living inside lipid membranes, forming the first simple cells. When this DNA reproduced itself, the new DNA molecule took some of the lipid membrane with it, to make a new cell. The cell would pull the membrane together in the middle, with one DNA molecule attached to the membrane on each side, and then pinch it off to make two new cells. Many prokaryote cells still reproduce just like that.
About two billion years ago, when these prokaryote cells evolved into the first eukaryote cells, the eukaryote cells began to have specialized parts inside them like vacuoles and the endoplasmic reticulum. It's more complicated for a eukaryote cell to reproduce because it has to also reproduce all of these specialized parts (organelles) inside it. Also, in order to make all these specialize parts, eukaryotes need a lot more instructions, so they have more than one molecule of DNA inside their nucleus. Eukaryotes have to make sure that all of their DNA gets reproduced and split up correctly.
Eukaryote cells like yeast usually divided through a process called mitosis, where the DNA of the cell reproduces itself and then lines up in the middle of the cell to systematically split down the middle. But by around 1.4 billion years ago, most single-celled eukaryotes could also reproduce in a more complicated way called meiosis. During meiosis, two cells each split into two smaller cells, but each with only half as much DNA as they need. Then two of the new smaller cells combined together again, making a new cell with half its DNA from one parent and half from the other parent. This method allowed the cells to mix and match DNA to make lots of slightly different cells, so that as conditions changed at least some of them would have a better chance of surviving. Meiosis greatly sped up the process of evolution, and by about 600 million years ago the first creatures with more than one cell formed.