Fundamental theorem of arithmetic (divisibility, primes, Euclidean algorithm, infinitude of primes)
Arithmetical functions and Dirichlet multiplication (Möbius function, Euler totient function, Dirichlet product of arithmetical functions, Dirichlet inverses and the Mobius inversion formula, Mangoldt function)
Congruences (Linear congruences, Fermat’s little theorem, Chinese remainder theorem, quadratic residues, quadratic reciprocity law)
Finite abelian groups and their characters (characters of finite abelian groups, orthogonality relations, Dirichlet characters)
Dirichlet theorem of primes in arithmetic progression (including the complete proof of non-vanishing of L(1,$\chi$) for non-principal $\chi$)
Binary quadratic forms (factorable and unfactorable forms, equivalence classes of forms, finiteness of class number of binary quadratic forms of a given discriminant)
Algebraic number theory (Algebraic numbers and Algebraic integers, ring of integers of quadratic extensions, quadratic reciprocity using Gauss sums).
Suggested books and references:
Apostol, T. M., Introduction to Analytic Number Theory, Springer International Student Edition, 1989.
Niven, I. and Zuckerman, H. S., An Introduction to the Theory of numbers, Wiley Eastern Limited, 1989.
Ireland, K. and Rosen, M., Classical Introduction to Modern Number Theory, Springer-Verlag (GTM), 1990.
Edmund Landau, Elementary Number theory, AMS Chelsea Publisshing, 1958.