The first idea that you might be able to share an internet connection with a neighbor is the fact that when you first setup your Wi-Fi system inside your house, you will often see that there are several networks of varying signal strength all listed on your computer. These networks listed are your neighbors, and quite logically when you notice that the signal strength of a neighbor's Wi-Fi connection is strong, it is fair to conclude that you probably could connect to their network, if you had their password, and hence be able to access the internet by piggybacking onto their network.
The way housing is structured in America, is that you have situations in which the housing is quite dense, such as apartments, condos, and townhomes, along with single family homes, to which typically the more windows a given dwelling has and the closer in proximity you are to each other, the easier it is to access your neighbor's internet and the more reliable and speedier that internet connection will be. While it is almost always true that a wired internet connection will be both more reliable and faster than wireless, a wireless connection is often acceptable for most people in their typical internet activities.
The main reason to consider sharing an internet connection with a neighbor is the fact that the cost of internet service can range from $30 to $70 a month, so the sharing of such, would equal to some considerable savings over a period of year, which as they say, "a penny saved is a penny earned". If you are fortunate and have a good relationship with a particular neighbor to begin with, it's certainly a reasonable subject to bring up, to which the objective should be to test out the theory first, without the need for anyone to get any additional equipment, by simply having one party providing to the other party, their password to their Wi-Fi and then taking it for a ride. Assuming that the test runs successfully, it then can be further discussed based on the results whether a Wi-Fi extender is something worth getting, or even installing a Wi-Fi antenna on your roof to make the signal much more powerful, all depending on the commitment neighbors have to one another.
Obviously, when there is sharing between neighbors of anything, there has to be general rules and responsibilities that each party must live up to, especially if your internet provider has a data limit, so that as long as there is a good understanding between neighbors, the savings for each neighbor more than makes up for any small inconveniences that must be overcome.
The bottom line is that there are many items that can successfully be shared neighbor to neighbor, and internet sharing is one of the more straightforward things to share of them all, as in most cases, you are simply sharing a password, and possibly equally the one-time expense of the modem/router and that is pretty much it. Remember that anytime that you log onto a public Wi-Fi network such as at a Starbucks or similar, you are sharing an internet connection, so in fact, internet sharing is done all the time; why not consider saving some dollars and doing the same thing at home.