DarthCaniac wrote that “Servers don’t connect directly to the internet”, which ironically enough is actually both right and wrong at the same time.
Your home network (either your PC, or your NAT router which provides connectivity to your other devices) is connected as directly to the Internet as most servers. Remember that the Internet is a network of networks. Your home network, if connected to the Internet, forms one small part of the entire Internet. On the Internet, your own network is, in a sense, like a drop of water in an ocean; nobody will really notice if a single drop is removed from the ocean, but if all drops were to be removed, then there would no longer be an ocean. The Internet is the same; lots of small networks all interconnected, any one of which can generally be removed with no disruption to the whole, but if you remove all of them then you have no Internet.
Most end-point sites are connected through a single upstream network provider. This goes also for many smaller businesses that for various reasons have hardware on-premises and have that hardware connected to the Internet, whether for the purposes of providing services to others or to simply allow their employees to browse Stack Exchange. This is your normal definition of an ISP; a company that provides you with connectivity to the Internet for your (host or) network, without any other special arrangements or expensive hardware required.
Some end-point sites are connected through a set of upstream network providers, but each is used as one would normally use a single upstream provider. This is often referred to as upstream connection bonding or multiplexing, and is a cheap way to get some degree of redundancy in your Internet service. More advanced small-business-class and up NAT routers have multiple WAN connections and are thus able to do this natively, or you can assemble something on your own using an old PC with a couple of network cards and some software magic. The primary difficulty in establishing this for an individual would probably be being able to obtain upstream service from more than one ISP simultaneously, as each ISP connection would likely require a separate physical cable (or other physical-layer link, such as a radio link), but it is by no means unattainable with some reasonable amount of money.
However, large end-point sites can use the same type of setup that those Internet service providers (that in the previous two examples you’d be connecting to) themselves use. Technically, what they do is known as peering with multiple upstream peers (or in some cases, simply peering with multiple peers in the parts of the Internet where the concept of “upstream” does not exist: in core routing, this is the default-free zone). This option is generally not available to individuals and it usually requires being willing to plunk down a fair amount of money on the table. At the very least, you will need to come to a “peering agreement” with at least two large ISPs in your area (you can do it with one, but that’s rather pointless except perhaps as a stepping stone), and in order to do so you will likely need a subscriber edge router (this is not the same thing as the home or small business NAT “routers” which are often referred to as just routers, but are really more like gateways than they are like the core routers of the Internet, and are in some contexts referred to as residential gateways even though they aren’t used only in residential settings) that can speak the Border Gateway Protocol, and in order to speak BGP you will need to apply for and receive an autonomous system (AS) number. You will also need to contact the Internet Registry in your region (RIPE, ARIN, APNIC, etc.), apply for and receive a globally routable block of IP addresses, and especially if you want IPv4, you are going to have to demonstrate your need for a sufficiently large block of addresses that people aren’t going to balk at having that in their routers, possibly even in the default-free zone, as well as demonstrate willingness to pay for the privilege of having those IP addresses assigned to you.
This last is probably what you meant to ask about, but as you can see, it is really quite involved. Additionally, unless you are a large company and/or providing Internet connectivity to others, there is really no significant advantage to it compared to the mid-end option of getting normal Internet access from two separate ISPs and multiplexing the upstream connections.