Internet addresses are allocated in large blocks to regional internet registries which cover large geographic areas (e.g. Europe is covered by RIPE). Each regional registry allocates smaller blocks to local registries, typically an Internet Service Provider (ISP). The local registry/ISP will assign smaller blocks or individual addresses to customers or dialup servers. Customers with small blocks then assign addresses to individual devices on their network.
As such, IP addresses are globally unique, and the job of your ISP is to ensure that any packet anywhere in the world sent to one of your addresses will reach your network. Similarly they ensure packets you send will reach the destination. They interconnect (peer) with other ISPs to exchange traffic destined for globally unique IP addresses. It is important to note that IP addresses do not belong to the end users, they are allocated to ISPs and assigned to users, but if you change ISP you will normally have to change IP addresses.