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.