1. Dismiss Promoted Tweets you don't like
Next to the icon showing that a tweet is promoted is "x Dismiss". Click the cross, and the Promoted Tweet will disappear from your timeline. One you've done this, you won't see the same tweet again, and it's less likely you will see Promoted Tweets in future.
2. Don't reply to a Promoted Tweet, EVER
Replying to a Promoted Tweet is interpreted by Twitter as a signal that you are interested in the tweet - even if you write something negative or complain. By replying, you make it more likely you will see Promoted Tweets from that advertiser and other advertisers in future.
Some Twitter users have very strong feelings about Promoted Tweets and make a point of replying to every one they see to register their displeasure, or protest that it's spam (note: Promoted Tweets aren't spam - find out why below.)
This user doesn't want to hear about flights:
@AirFranceUK spam— man united (@manutdreserves) September 11, 2014
Or about marketing technology:
@atlassian spam. Blocked— man united (@manutdreserves) September 10, 2014
Or about furniture:
@HarveysHQ if I wanted that stuff I'm capable of looking for it without the help of your spam. Blocked— man united (@manutdreserves) September 10, 2014
Ironically, replying has the opposite effect of what they intend. Twitter thinks they really like seeing Promoted Tweets and therefore shows them as many as possible.
3. Avoid clicking on anything in the Promoted Tweet (except the "x Dismiss")
Other actions, like expanding the tweet to see the replies, or clicking on the profile picture or handle of the advertiser are also interpreted by Twitter as signs you like the advert. As such, clicks make it more likely you'll see more Promoted Tweets in future.
Why can't you just not show your ads to me?
Twitter advertisers don't choose which individual users see their ads - that's done by Twitter, based on criteria the advertiser gives them. For example, at Bought By Many, if we want people to know about our group for Rescue Dogs, we might ask Twitter to show our ads to people who follow animal shelters and rescue centres. Or, to reach Young Drivers, we might ask Twitter for an audience of people who have used words like "black box car insurance" in recent tweets.
You can read more about how Twitter decides what ads to show you here.
Won't Twitter just give up on Promoted Tweets if users complain enough?
I'd say that's unlikely. Twitter is free to use, but costs hundreds of millions of pounds to run. Advertising is its primary source of income, and its investors expect it to be making more and more money from ads as time passes. Over time, though, Twitter's algorithms will get better at determining which ads to show to which users.
What's the difference between Promoted Tweets and Twitter spam?
Promoted Tweets are paid for by advertisers. They can only come from bona fide Twitter accounts and Twitter vets all prospective advertisers before it allows them to advertise. It also monitors their ads and stops ads that don't comply with regulations or are inappropriate or offensive.
By contrast, Twitter spam involves accounts that have either been hacked on set up specifically to send unsolicited messages. These accounts then tweet other users indiscriminately, or with crude targeting (use the word "fat" in a tweet and you might get an automatic reply from a spambot with a link to a weight loss product). These accounts haven't been vetted or approved.