What is the difference between a rollover and a trustee-to-trustee transfer?

A rollover is a tax free distribution (withdrawal) of your HSA funds from one provider to another. The funds are typically distributed to you (you take possession) and then you have 60 days to deposit those funds with a new custodian. You are limited to one HSA rollover per year (every 12 months). 

A trustee-to-trustee transfer is when HSA funds are transferred directly from one custodian to another (you never take  possession of the funds). There is no limit on the number of transfers you can make from one HSA account that you own to another HSA account that you own.

Both methods are tax free so long as all of the rules are followed.