You only have a few seconds to make a positive first impression, so when it comes to meeting your new boss for the first time, you'd better do your best to start your working relationship off on the right foot. Like any other person, your boss is going to be sizing up your clothing, attitude and general demeanor.
If he's your new boss, it means you've already gone through the interview process and been hired. Still, that time you spent interviewing probably gave you a sense of what it's like to work for the company, what other employees wear, and the general level of formality around the office. All that information is going to serve you well when meeting your boss for the first time and give you lots of clues about how to behave. If your boss is the one who's new, expect her to know something about the company culture, but don't expect her to automatically adopt the same level of formality as previous bosses.
First off, your boss is going to notice how you're dressed and how well you groom yourself. In some more formal offices, not being well-groomed is going to tell your boss that you're less than professional. If the company has a "business" or "business casual" dress code, take it to heart and err toward more formal business attire. Shine your shoes. Wear a few key, classy accessories. Get a manicure and a haircut before you meet the new boss.
How you behave also matters. When you meet the boss for the first time, maintain an open, friendly demeanor. Shake her hand in a confident manner, but don't go for the hand-crushing, overzealous handshake. Look her in the eye and relax your posture. As you walk around the office or when you're sitting in a chair with the new boss, avoid crossing your arms. Doing so signals that you're not open to what the boss is saying, or that you're closed off or nervous.
Addressing the Boss
There's also the question of how you'll address your new boss. The key here may be to follow the boss' lead. Address the new boss as "Mr." for men or "Ms." for women, unless you know for a fact that she's married. Once you've addressed him once in this way, he might ask you to call him by his first name. Also pay attention to how other staff members address him. If he's consistently addressed as "Dr. X," or by some other formal title, follow your co-workers' leads.