As a Senior Engineer...
An organization's cohort of senior engineers are the ones that get stuff done. They design and implement systems to solve the hard and important problems, and mentor and act as role models to other engineers.
You'll be working on designing systems, teasing out complexity, breaking tasks down for more junior engineers to sink their teeth into.
You will start to become 'the' person who knows technically the most about your systems. Managers will look to you for advice.
Senior engineers should strive to look out for their system's future well being, and act as a counter-balance to management's and business' desire to implement something as quickly as possible.
That being said, you should understand that the goal of a software engineer in an organization is to enable the company to meet their business goals; software engineering and developing systems is a means to this end. Don't forget that business revenue is what pays you and your team.
If you have concerns about your system performance, make sure to make it known to your manager. It won't hurt to include performance and monetary impact if not addressed, if you have those figures.
On progressing past senior
There isn't an expectation that individuals at the senior level will need to develop further to hit the next rung on the ladder: 'staff' (or principal, or tech lead, etc. there isn't a standardized role label for the ones past senior engineer...).
There also isn't an expectation that the senior developers will need to transition into a management role either (though I think most companies only offer this state transition starting at the senior level).