So I'll recommend LastPass. It meets all of your needs:
- Windows Compatible (7): Yes. Cross-platform (Windows, Mac & Linux) and cross-browser compatible (all 5 common browsers)!
- Remotely accessible from my android device (HTC1): Premium version only AFAIK (haven't used it on any remote devices). (there are apps for all the major smart phone OSes)
- Very Secure. I want to use it for Banking and my Work System passwords (I don't know much about security if anyone can elaborate/suggest an expected standard for me): Lastpass is very secure; of course generally it is only as secure as your master password. However you can enable multi-factor authentication (might need premium I can't remember - since I don't have cell signal usually I haven't bothered to set that up for anything despite the benefits).
- Cost is relevant, <100 USD, preferably one time but annually would be fine as well. Cost is secondary to security and functionality, so do not hold back a recommendation for cost concerns. Free version available. Premium version is well within your price range - $12/year.
Lastpass is also more than just a password app; you can securely (ie encryptedly) store important documents and notes and credit card/address details as well.