What makes a magnet on the atomic level?

Magnetism at atomic /sub - atomic levels is mostly due to charged particles called electrons. Electrons have spin which give them angular momentum and thus a magnetic moment associated with it. That is the cause of Magnetism at atomic levels. Electrons fill up orbitals in atoms in pairs.

adding a(n) to an electric circuit would increase the magnetic field around a wire;

b. solenoid

a solenoid is a type of electromagnet whose purpose is to generate a controlled magnetic field.

if the purpose of the solenoid is instead to impede changes in the electric current, a solenoid can be more specifically classified as an inductor rather than an electromagnet.