Each of those is a network. Merchants displaying their logos - participate in their network and will accept cards that bear the same logo. Most merchants participate in more than one network.
Discover is mostly used in the US, while Visa, Mastercard and American Express are more widely spread in the world (Amex less, Visa and MC are much more widely spread).
In addition to being widely spread in the US, Discover is accepted everywhere where UnionPay is accepted (mostly in China) and Diners Club (mostly in EMEA).
Advantages/disadvantages? You'll have to compare specific cards, but if you're a traveler in the world - then Discover will probably not be as appealing as Visa or Mastercard.