How to... Internet without ISP?
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.
What does "connected to the Internet" really mean anyway?
It really means "has a route available to send packets to all or virtually all devices on the Internet and a corresponding route to get replies back".
To do that, ISPs and other large networks interconnect with each other. There are two main types of interconnections, "peering" and "transit" (and a few variations between these two).
"Peering" interconnections are usually but not always "settlement free" (neither network pays the other for the peering although there does need to be some arrangement to pay for the physical connection) and allow the two networks and their customers to communicate with each other. Peering is not transitive. If A peers with B, and B peers with C then A doesn't get a route to C.
Peering may happen either on a direct link between the two networks or via exchange points. Exchange points provide an interconnection (nearly always an Ethernet network) between many providers, making peering more viable between providers who only exchange a small amount of traffic. If two providers are exchanging a lot of traffic (sustained gigabits per second nowadays), then it's usually more economical to put in a direct link.
On a "transit" interconnection, there is a provider-customer relationship. The transit provider provides (for a fee) the transit customer with connectivity to the entire internet. A customer may use multiple transit providers for redundancy or to provide shorter routes.
There are a handful of big providers known as "Tier 1" networks. These networks don't buy transit from anyone, instead they all peer with each other. Becoming a Tier 1 network is extremely difficult because you have to convince all the existing Tier 1 networks to peer with you and most of them are pretty reluctant to take on new peers.
Large networks that are not Tier 1 will try and push as much of their traffic as possible onto peering relationships (because it's cheaper and usually faster). However, they will have to buy some transit to get to networks where they can't establish a suitable peering relationship. Many large ISPs also avoid peering with little guys, either because they see those little guys as potential customers, because the administrative overhead is not worth the benefits, or because it would mean they'd effectively be giving away international transport for free.
According to CAIDA (which is not always 100% accurate - in particular it misses many peering connections), Google's main network has 3 providers, 10 "customers" (all but two of which are other Google networks), and over two hundred peers. http://as-rank.caida.org/?mode0=as-info&mode1=as-table&as=15169&n=227&table-details=simple
Playing the peering game costs money. You need to get your own IP space and AS number which will involve paying fees to a RIR and is further complicated by the exhaustion of IPv4 space for regular allocations. You need to buy routers that can import the whole internet routing table. You need to rent ports on the exchange points you plan to use and links from your rack space to the exchange point. If you use direct links for peering, you will need to buy or rent those. You will need to have network operations staff available 24/7 (most providers insist on this as a precondition for peering). Also, you will find many providers don't want to be bothered peering with you unless you can demonstrate that there will be significant traffic flow.
If you want your international traffic to run over peering, then you will likely need to have a presence at many exchange points in both the US and Europe. This will involve renting connections to carry that traffic.
The result is, unless you have a LOT of traffic (nowadays probably gigabits per second sustained), it's cheaper to just buy a connection from an ISP and let them worry about playing the peering game.
By definition, ISP means "Internet Service Provider". If a server gets its internet from the datacenter, the datacenter is the ISP.
Servers don't connect "directly" to the internet because there is no single company that owns the internet. There are many large companies such as XO Communications and Level 3, that own so much cable backbone throughout the world that people pay them for access. Those networks share data amongst themselves usually at no cost, but unless you own an entire countries worth of fiber optic lines, you probably won't get a free internet connection.