An allele goes to fixation (frequency = 100%) or to loss = extinction (frequency = 0%). This may happen in populations of any size (except 0).
Alleles go to fixation or loss because of the action of evolutionary forces.
Alleles may also go extinct because of extinction of populations in which they occur. [Or not — e.g., Evolutionary Rescue]
A population persists (at some size) or becomes extinct (population size=0). Genes in populations may be polymorphic (two or more alleles per locus) or fixed for any one allele (the other alleles are lost, i.e., have become extinct).
Populations may go extinct* due to a wide range of factors; the proximate causes are ecological.
*extinction of any one population of a species is termed “extirpation”
Here is a great summary of a very complex set of interactions