When we talk about alleles being "dominant" or "recessive", we really are just talking about the variations of the nucleotide sequences of the organism's DNA because that's what the definition of an #color(blue)"allele"# is.
Say, for example, we are studying the gene that determines seed shape in pea plants like Mendel studied in his famous experiments.
#color(white)(aaaaaaaaaaaaaaa)ul"Seed shape in pea plants"#
Figure 1: The left pea seed is round (dominant) and the right pea seed is wrinkled (recessive).
The #color(red)"dominant allele (R)"# (again, we are talking about the nucleotide sequence of the DNA) codes for a certain enzyme that helps to convert an unbranched form of starch to a branched form.
The #color(magenta)"recessive allele (r)"# codes for a defective enzyme, which leads to a buildup of the starch in the unbranched form due to unsuccessful catalysis. This in turn causes excess water to enter the seed via osmosis so that when the seed dries up, it will wrinkle, leading to the appearance of a wrinkled seed shape.
So let's take a pea plant which is heterozygote for seed shape. Knowing that there are #"2 chromosomes"#, where one chromosome has the #color(red)"dominant allele (R)"#, and the other chromosome has the #color(magenta)"recessive allele (r)"#, together they give a heterozygote genotype for the seed shape as #(color(red)"R"color(magenta)"r")#. But because we have #"1 dominant allele"#, it codes for just enough enzyme that will help to catalyze the reaction of the unbranched form of starch turning into the branched form, leading to the round-shaped seed appearance. This would be no different from a seed that expressed BOTH dominant alleles, #color(red)"(RR)"#.
Note: The pea plant that shows a genotype of #color(magenta)"rr"# would NOT be able to produce any functional enzymes due to both chromosomes coding for defective enzymes, meaning, the starch would be left in the unbranched form. That is why you would see the wrinkled seed shape appearance.
Figure 2: One dominant allele form is enough to lead to a round shaped seed
Source: http://lbc.msu.edu/evo-ed/Pages/peas/genetics.html with editing done on my part.