How can I save information locally in my chrome extension?

You can now leverage Google Chrome's storage API to do this as well. Unlike localStorage, this is accessible from content scripts as well.

//  Somewhere in your manifest...

{
    "permissions": [
        "storage"
    ]
}

//  Usage:

//  PERSISTENT Storage - Globally
//  Save data to storage across their browsers...

chrome.storage.sync.set({ "yourBody": "myBody" }, function(){
    //  A data saved callback omg so fancy
});

chrome.storage.sync.get(/* String or Array */["yourBody"], function(items){
    //  items = [ { "yourBody": "myBody" } ]
});

//  LOCAL Storage

// Save data to storage locally, in just this browser...

chrome.storage.local.set({ "phasersTo": "awesome" }, function(){
    //  Data's been saved boys and girls, go on home
});

chrome.storage.local.get(/* String or Array */["phasersTo"], function(items){
    //  items = [ { "phasersTo": "awesome" } ]
});

More info on how these shenanigans work here: https://developer.chrome.com/extensions/storage#type-StorageArea

Former answer:

Use localStorage. Google Chrome implements some features of HTML5, and it is one of them.

//Pull text from user inputbox
var data = document.getElementById("this_input").value;
//Save it to the localStorage variable which will always remember what you store in it
localStorage["inputText"] = data; 

You should note you can only access your storage from the background page (no content scripts) so you'll have to use messaging for that.

There is a specific API now, check this here: https://developer.chrome.com/extensions/storage

Note that it is only available on your background page and works in an isolated world (you don't access the values contained in the HTML5 localStorage of the web site).