How can I get a different seat after boarding a flight?

You do not have much options, but you can always try one of the following:

• Simply move to an empty seat: After boarding is done and doors are closed, move to any empty seat within the same class. Flight attendants will not mind.
• Ask other passengers to swap: Be smart and ask people who are not with family or friends so it will not be a problem for them to swap seats. Older people hate window seats, younger people love window seats, ask the right people with a good smile for the seat and they will swap.
• Talk to the cabin crew: They are always willing to help (if they can). Just tell them your reason and be honest, you can even say something like "the guy next to me smells bad!", or "I am planning to sleep and I do not prefer sitting next to babies", of course say that in private not in public, and they will help you as long as they can.

Some more info on top of what @MeNoTalk said:

It mostly depends how full the airplane is and why you want to switch. If the airplane is rather empty, there is no issue, you do not need a reason and you can switch as you like. If it is full, it depends how personal the reason is and if another (higher or lower) class is still free or if it is acceptable to get someone else into the seat.

Examples:

• If your seat is broken, they will not put someone else there, same if it is next to a baby etc.
• If you are a woman and can argue that you feel uncomfortable sitting next to a specific man, they will most likely re-seat you.
• If it is about sitting next to your friend/spouse, you can simply go and find someone single to switch with you or your friend. Make sure that the other person switches to the better seat (towards the front etc)
• If the staff is very busy, you likely better help yourself and find someone who would switch with you.

General rules:

• If you see the issue right away, ask immediately during the boarding procedure if the airplane is full. If it is not, either sit down or stand at an emergency exit and wait for the doors to close. Make sure you get your hand luggage into a bin already however.
• Discretion and politeness is 99% for a good switch. Talk to the staff in private (as mentioned) and indicate that you are fine to switch later, too. So you take the pressure off them. Try to indicate your wish to switch as your problem/weakness instead of a demand towards the staff.

A few quick comments:

• I've had people who were seated beside me get up and go sit somewhere else. It's not that weird, and nobody gets offended.
• IMHO, you should NOT get up and go sit beside someone else you don't know (without a really good reason). That other person was about to enjoy a comfortable flight with some extra room, and you would be taking that away (basically saying your marginal increase in comfort is worth a significant decrease in their comfort - that's not right).
• If your goal is to sit next to the pretty girl, then you shouldn't be doing that.

For the longest while, prior to the carry-on-luggage madness of today, I would make a point of being the very last person to board the airplane. That way I could walk down the aisle and literally pick any seat I want, knowing that nobody else would show up to claim it.

...and by the way, it isn't the end of the world if you claim a seat and someone else shows up who actually has it assigned. That's happened to me too, and I just admit I was hoping to snag a better seat and surrender it to them. Nobody cares.

Just move! I've never had a problem moving to an empty seat on a plane. Normally this works best when the plane is half-empty anyway and there's an entire empty row. Just remember to wait until everyone else is boarded (so you don't annoy anybody) - very shortly after the doors are shut is a good time - and be quick, as there's often only a short period before the plane pushes back. I've also switched seats after takeoff (and the sign is switched off), and again, I've never had a problem with that.

Don't just move! Extremely bad advice. Always involve the crew when moving to a different seat.

1. Some flights have weight distribution concerns and certain seats will be blocked off. You have no way of knowing this as there will be no signs, simply those seats won't be assigned. And it's not something that can be handled later, the weight distribution is most important at take off time.

2. The manifest on most domestic flights and definitely all international flights has information on who sits where. This can be important after a catastrophe and theoretically the crew will update it if you swap.