No, Beats by Dr. Dre headphones aren't and never were made by Apple. Apple sells Beats by Dr. Dre headphones but that is all. Monster used to manufacture Beats by Dr. Dre headphones but recently their five year contract ended and Beats by Dr. Dre is making their headphones independently.
iPod earphones are compatible with most standard headphones and earbuds, which are readily available just about anywhere. Apple brand earbuds can be purchased from the online or storefront Apple Store, or from reseller sites like eBay.
Amazon, one of the most well known retailers in America, has a very large selection of headphones to be used with the Skype service. Options for Skype compatible headphones include those that work wirelessly as well as hands free.