Update: There is a very helpful catalogue of all the rules developing over on reddit.
As with all things, your mileage may vary in applying for cards. There are three commonly applied rules that you need to know about when you start travel hacking. You'll need to keep these in mind as you start applying for cards.
1. Chase 5/24 Rule
Chase 5/24 Rule is among the first rules people learn about travel hacking, and unfortunately many (including me) learn it too late. The basics of the rule are as follows. Chase will not approve you for a credit card if you have 5 new cards (any cards, not just Chase) opened in the last 24 months. The 5/24 rule does not apply to all cards. That is, Chase will give you some cards even if you have 5 new cards in the past 24 months.
The 5/24 rule got some attention when the Chase Sapphire Reserve was released because many churners and travel hackers had more than five cards opened in the last 24 months. This is why we point out that using sign-up bonuses often works better for people who travel infrequently.
(Chase also has a 1/24 rule, which is that you can only get a signup bonus for given card once every 24 months. This is actually the most generous policy regarding multiple signup bonuses among the three major issuers.)
What it means for you
Because Chase has an impressive portfolio of Chase-branded and co-branded cards, the 5/24 rule is the first one most new travel hackers need to keep in mind. Many articles on cards divide the universe into "if you're under 5/24" and "if you're over 5/24."
2. Citi Family 1/24 Rule
Citi's 1/24 rule is that you will only get a signup bonus for a card if you have not gotten a signup bonus for a card in that same card family within 24 months. Doctor of Credit has a list of sample language for Citi's different cards. Update: Here's some more from Lucky over at One Mile at a Time on Citi's rules.
Note the difference between this rule and Chase's 1/24 rule. I was able to get the signup bonus on the Chase Sapphire Reserve despite having opened the Chase Sapphire Preferred within the past 24 months. To get the bonus on the preferred again, however, I have to wait the full 24 months. Conversely, I opened the Citi Prestige in the last 24 months, so I cannot get a sign up bonus for the Citi ThankYou Preferred until the 24 months are up. I could, however, get a signup bonus for any of the Citi Aadvantage cards, since they are in a different card family.
Also, unlike Chase 5/24, you can still open a Citi card in the same family, you just won't be able to get the signup bonus.
What it Means for You
First, you have to understand the Citi card families. Second, since you won't get a signup bonus with a new card, you'll want to pick the card that most fits your lifestyle. This is why I don't have an Aadvantage card yet - I can't choose among the options. Right now, they all have great sign up bonuses (50,000 miles), but since I'll only get those from one card, I want to choose right.
3. Amex Once in a Lifetime
Among the harsher rules is the American Express once-in-a-lifetime rule. You can only get a signup bonus for each card once in your lifetime. Note that like the Citi 1/24 and unlike Chase 5/24, you can still open a card covered by the rule, you just won't get the signup bonus.
The harshness of this rule is somewhat mitigated by the variety of Amex platinum cards around (for example, the Mercedes Benz Platinum Amex).
What it Means for You
If you're new to travel hacking, you'll want to weigh how much you need American Express Membership Rewards points, or other card perks, now versus later. You can get targeted American Express offers, too. I've seen 75,000 points in the mail within the past year and heard of 100,000 point offers online. I'll be waiting for 100,000 before diving into the pricey American Express Platinum, for example.