My Own Words
Office365 is a Microsoft Hosted Cloud based solutions that allows small to medium sized companies to quickly setup and use the Microsoft Information Worker Technologies (SharePoint, Exchange, Lynch, Project and Microsoft Office). Office365 allows these organization to save cost of servers, maintenance, support. It works great for people on the go and is available on every platform (PC, Mobile and Web). You are connected all the time. SharePoint Online is one of the services of Office 365. Microsoft Maintains Office365 and data is placed geographically dispersed data centers that allows you to be highly available with 99.99% confirmed uptime. There are different plans for Office365. E1, E1, P1, P2. These plans are based on company size like 8 to 10 People company, 1-25 sized company, 25-100 to more. You pay different prices for different plans e.g. P1 is 4 USD Per Person Per Month. P2 and 8 USD and go on. for more information you can use http://office365.com/ or http://office.microsoft.com/en-us/business/?WT%2Eintid1=ODC%5FENUS%5FFX101785584%5FXT104052692&WT%2Eintid2=ODC%5FENUS%5FFX101785584%5FXT104029222
Microsoft has recently upgraded Offce365 to 2013 version of SharePoint, Lynch, Exchange and Project. With Respect to Integration of SharePoint on premises with SharePoint online. it is good for organization which has lot of people on the move or external users and they don’t have access to Organization so you can setup a "Hybrid" deployment. You can use Sandbox Solutions to bring data in from Office365 based SharePoint online sites. You can not develop any Server Side apps for SharePoint online.