Note: An saol refers to the period of time covered by your life, i.e. "It was here my whole life".
Beatha is used to refer to life in other circumstances, such as "quality of life, my way of life".
Finally "Anam" is the animating quality of life which distinguishes animals and humans from a stone.
Saol has a derived meaning of "The world, All people".
Anam has a derived meaning of "the immortal soul".
Anam would be the norm, "Chaill sé a anam" = He died, "Gan anam" = lifeless.
Beatha is sort of "life" as in your wealth, job, way of conducting yourself.
Now, non-native speakers will probably write "Comhartha beatha" for "life signs", but native books will usually convey it via some phrase with anam.
I should also emphasise that "Anam" is specifically the animating force, so plants are "gan anam", although they are "beo", i.e. alive. There isn't really a native noun for "alive" in the general sense.
Although probably younger speakers would consider plants to have "Anam", I know one from Donegal who told me this specifically and I know a few native speakers who use "Anam" for Qi/Chi.