What is the abelian variety?

Excuse me, could you please elaborate on the concept of an abelian variety? I understand it's a mathematical object, but I'm having trouble grasping its specific definition and characteristics. How does it differ from other algebraic varieties, and what makes it unique in the realm of algebraic geometry? Additionally, are there any real-world applications or implications of studying abelian varieties that I should be aware of? Thank you in advance for your clarification.