This would entirely depend upon the context.

Both 'anpartselskab' and 'aktieselskab' are Limited companies and so if you are actually translating the word you need to distinguish. An 'anpartsselskab' is smaller than an 'aktieselskab' and does not need so much captial to set up.

The answers received already pretty much covers it, but for exact information on each type of company, please refer to 'Danmarksserveren' and the link below.

My own preferences are

aktieselskab: public limited company
anpartsselskab: private limited company

Eurodicautom defines the two company forms as follows:

Aktieselskab:
a company whose capital is divided into shares which can be offered to the general public and whose members are only liable for its debts to the extent of any amount unpaid on their shares;
the company has the status of a legal persona

anpartsselskab:
a company whose capital is divided into shares which cannot be offered to the general public.The liability of its members is limited to the amount of their shares.The private limited company has the status of a legal persona

These are the possibilities suggested by my Business Dictionary:

Aktieselskab:
[selskab] company, corporation (US), incorporated enterprise, joint-stock company (UK), limited company, limited liability company, public corporation (US), public limited (liability) company (PLC) (UK), stock company, stock corporation (US)

anpartsselskab(ApS):
limited liability company,
private company (Co. Ltd.), private limited (liability) company (Co. Ltd.); (australsk/sydafrikansk) Proprietary Limited (Pty. Ltd.) - Euodicautom adds limited-liability partnership