If a homozygous dominant brown mouse is crossed with a heterozygous brown mouse what will the offspring be?

Time for a Punnett's square! If we assume simple Mendelian genetics (no codominance, no interaction between genes, only 2 alleles, etc)

B = Dominant allele
b = recessive allele.

Homozygous dom brown mouse = BB
Heterzygous brown mouse = Bb
---B--|--b
B|BB|Bb
B|BB|Bb

As a result, you will get 50/50 Homozygous dom brown mice to heterozygous brown mice.

Let B = dominaly allele (brown)
Let b = recessive allele (tan)

An individual who is homozygous dominant carries two alleles that code for a dominant trait.  In this case, the homozygous dominant can be represented as BB

An individual who is heterozygous carries one dominant allele and one recessive allele.  The dominant allele is expressed in the phenotype.  In this case, the heterozygote can be represented as Bb

Hence, your cross is as follows:

BB x Bb

Based on this punnet square, you can conclude the following genotypes and phenotypes:

GENOTYPES:
50% of the offspring will be homozygous dominant (BB)
50% of the offspring will be heterozygous (Bb)

PHENOTYPES:
100% of the offspring will be brown

Hybrid.