Being a Summer Intern at a startup myself, I hope I can bring a couple of things to the table.
First things first...
Learn your company's environment(like flexible/perfectionists), start digging their official background(like business process, connection with other companies). They might help you at certain points of time(when you don't want yourself to feel odd among the colleagues).
You should be good with grasping the concepts easily, as they don't want to end up the whole day make you understand things. You should learn to be on your own. You are all by yourself. You should prove yourself that you are capable of taking part in the deciding factors of a company.
You should be bold enough to point out the mistakes if they do any, but only when you are sure about it and when you can strongly back up your point with reliable reference/resource.
You will be highly expected to get yourself involved in the company's business process, that will make them think its worth of hiring you.
They need you to be very active, passionate enough.
Employers expect you to be not just straight, but flexible too wherever/whenever needed.
Also you would be expected to have personal experiences on programming or whatsoever it concerns.
Keep your schedule. Always mind your words as they are the key factors to judge you.
To summarize, you should be worthy of being hired.