- 'Entry Clearance' is just another name for a visa issued outside the UK - in your case, it just means that it gives you permission to enter the UK for the purpose of employment (essentially leave to enter).
- 'Leave to Remain' another name for a visa issued inside the UK (because you are already in the UK on a different/expiring visa and are being given permission to remain there on a new visa). You would only ever need to apply for 'leave to remain' if your Tier 2 visa expired and you needed to renew it, or if you were switching from a Tier 2 visa to a different type of visa while you were living in the UK.
So, in terms of your Tier 2 visa, you will apply for it in the US (along with any dependent visas), you can pay $150 per application for 48-hour processing (purchased from the Worldbridge website), and then when it has been issued, you move to the UK... that's it.