Single page apps are those that live on a single HTML document. This means that if you want to display some different content to the user, depending on the state of the application, you will need to do some DOM manipulation (cutting out and replacing certain elements of the current document with different HTML) in order to update the 'view' that the user sees. Excuse me if this is obvious to you, please don't take offense. I figured I'd start from here. Hang with me and I'll explain how your routing situation is going to play out (more or less).
URLs are composed of a few different parts, each of which informs the browser of a particular bit of information that is required in order to download the resource that the user is attempting to access. Typically the resources that you are looking for are off on a server somewhere and the browser knows this because of pieces in the URL like 'protocol' ('http:') and 'host' ('www.mydomain.com'), so it goes off to that server to find what you're requesting. There are also 'query' parameters in URLs which provide some additional information to the server regarding a particular action, like the search terms of a search query. After the query parameters, comes the 'hash'. The hash is where the magic of single page apps happens... eh, well, kind of.....
First a bit about the hash. When you add a '#' to a URL, the browser then interprets the information that comes after it to be some location (element) within the currently displayed document. That means, if you have an element with an 'id' of 'main' and you add '#main' to the end of the URL, like so: 'http: //www.example.com#main', the browser will 'scroll' (typically 'jump') to the beginning of that element, so that the you can see it. Be aware, though, that if you type 'http://www.example.com/#main' (with the hash separated from the URL by a slash) then you will force a complete page reload and the browser will attempt to find a file by the name '#main' on the server (I bet it doesn't find it).
The takeaway here is that the browser will not attempt to navigate away from the current document if there is a hash in the URL
single-page apps don't want to navigate away from the page
Now, this whole thing about the hash isn't vital to single-page apps, as you could make one without dealing with it all. A bunch of click handlers and DOM manipulation is all you'd need really... But, that would mean that users will have no way of sharing links to particular views in your app. The URL would never change, and we would never be able to navigate to any particular view directly. We'd always be starting from the starting position of your app, which could easily be a very annoying situation.
If your single-page app is going to have different views, and you want users to be able to navigate directly to particular ones via bookmarks or links, then you will need to implement a form of routing on the front-end in addition to the routing that you'll need to implement on the backend (routing for data API, etc.), which means that you will need to make use of the hash.
I don't want to get into how different frameworks accomplish routing on the front-end, but it's basically a matter of updating the browser's address field when the user clicks a link, and watching the address bar to determine what the current URL is and loading the HTML that is associated with that URL into the DOM in the designated location in the document tree.
So, within a single-page app, you have one route on the server that deals with rendering the app HTML document (index.html), and you have routes that are responsible for dealing with the data of your app (creating new instances in the database, logging in and out, editing or destroying instances in the DB, and fetching data...) which are called via AJAX requests.
This is actually a fairly complicated situation in that HTML5 allows us to be able to forgo the hash (with the help of some link rewriting on the server) and also be able to use the 'back' and 'forward' buttons as if we've actually navigated away from the original document (which we haven't because we have only pointed the browser to the exact same URL, only with modified hash values, so no new page loads have occurred). Traditional site navigation and linking can be achieved by utilizing the browser's History API, which is available for IE beginning with version 10 (I believe), the rest of the big browser vendors were already on to it quite a bit earlier, so frameworks that leverage this technology will allow your users to navigate your app without the hash in the URL. Explaining this is a diversion and not necessary for understanding routing in single-page apps, but it is interesting and you'll have to learn it eventually anyway, probably..
AJAX should be used to request JSON from the server. AJAX requests will always hit your server because you don't include the hash symbol in AJAX requests (it would be ridiculous to do so because the hash is meant only for in-document browsing), so server-side routes must be responsible for exposing your data API (consider a RESTful one). While this is not their sole purpose in single-page apps, it is perhaps their most important one.
EDIT A quick mention of AJAX implementation.
On the server, you will have routes similar those that Alex has provided as examples and you will make calls to those URLs from the client using whatever XMLHttpRequest (XHR) object is exposed by your framework or library of choice. It is now considered more or less standard and best practice for frameworks/libraries to implement these requests as Promises http://wiki.commonjs.org/wiki/Promises/A. You should read a bit about it on your own, but I might be able to summarize it by saying that it is an asynchronous operation analogous to 'try, catch, throw' in synchronous operations. You will instantiate a promise object and through it you will attempt to load data from the server, for instance, via GET request. Make sure that you have assigned functions to handle requests made to the URL that you made the request to (server-side route)! This object that you instantiate and subsequently make the request to the server through, promises to return the result of the request to you once it comes back from the server (no matter whether it was successful or not) If it is successful, it will call a function that you have written and will supply it with the data from the server. If it fails, it will call a different function, also written by you, and will supply it with the error object (or 'reason' for failure), so you can handle the error appropriately.
whoa, thank you so much! the time you took to explain it soooo well. Thank you, it does help for sure. BTW, just yesterday I was watching a great presentation of promise/A+. Highly useful youtube.com/watch?v=hf1T_AONQJU
Nice, I'm glad it could help. This was my first time actually contributing to SO, which is probably a bit messed up considering how often I am here looking for answers... What framework are you using to build the front-end?