I've written a lot in past about using credit card points to purchase "free" travel and how to decide if a premium travel credit card makes sense (and choosing the best one for you). There are really two ways for travel-oriented credit card holders to use their points towards free travel:

Use those points like cash to purchase airline tickets directly through the credit card issuer, or
Transfer those points to your airline account so that you can purchase tickets with miles.

Note that all of my examples in this post will use economy seating so we can compare apples to apples. We'll cover the nuances of first and business class in a later post.

Basic Math

There's some basic math behind how this works. When you use your credit points like cash to purchase tickets directly through the card issuer, each point ends up being worth a very specific dollar (or cents) value depending on which card you have. Examples below.

Chase Sapphire Reserve cardholders get 1.5 cents for every point spent on air travel. That means a $100 plane ticket is going to run you about 6,667 Chase Ultimate Rewards points because $100 / .015 = 6,667.
Citi Prestige cardholders get 1.25 cents for every point spent on air travel (or at least they will starting in 2017... until then they get a bit more). So that same $100 plane ticket will put you out $100 / .0125 = 8,000 Citi ThankYou points.
American Express Platinum cardholders get 1 cent for every point spent on air travel. So, again, math... the $100 ticket will cost you $100 / .01 = 10,000 AMEX Rewards points.

Conversely, the mile rewards programs of many airlines use more of a zone-based scale for free travel. In other words, most one-way economy tickets between the United States and Europe are going to run you 30,000 miles / points on United (30k is a fairly standard benchmark for the US-EU route).

Dance Partners

Each of the major credit card brands offer a scheme whereby you can transfer your points from the card to their airline partners. Each of these agreements are negotiated separately, so they vary from card to card. Examples below.

Chase Sapphire Reserve cardholders can transfer their points to British Airways, Air France / KLM, Korean Air, Singapore Airlines, Southwest, United Airlines, and Virgin Atlantic.
Citi Prestige cardholders can transfer their points to jetBlue, Eva Air, Etihad, Air France / KLM, Garuda Indonesia, Malaysia Airlines, Quantas, Qatar, Singapore Airlines, Thai Airways, Virgin America, and Virgin Atlantic.
AMEX Platinum cardholders can transfer their points to: AeroMexico, Air Canada, Air France / KLM, Alitalia, ANA, British Airways, Delta, El Al, Emirates, Etihad, Hawaiian, Iberia, jetBlue, Singapore Airlines, Virgin America, and Virgin Atlantic.

So let's put the pieces together...

When it makes sense to use points like cash

In general, the cheaper the dollar value of the ticket, that is, the cost that you'd pay for the same ticket in dollars, the more advantageous it is to use your credit card points like cash to purchase your ticket directly through the card issuer. Also, in general, Chase members will get a better bang for their buck than will Citi members, who will in turn do better than AMEX members. Let's look at some examples...

A one-way United ticket to Europe is on sale for $400. It makes more sense to buy it outright for 26,667 Chase points than to transfer 30,000 points to United to purchase the same ticket.
Using the example above, let's say the $400 ticket is on Delta, and you're an AMEX customer. It makes no sense to purchase it outright for 40,000 points when you can transfer 30,000 to Delta and buy the ticket through the airline.
Thinking domestically, let's say you're a Citi cardholder considering a $100 ticket from DC to Boston on American. You could spend 8,000 points to use your points like cash ($100 / .0125), or you could transfer 12,500 points to American and buy the ticket with miles.

Overall, the cheaper the ticket price, the better idea it is to use points like cash. In general, this strategy will be most effective on domestic flights.

When it makes sense to transfer to your airline miles account

In general, the more expensive the dollar value of the ticket, that is, the cost that you'd pay for the same ticket in dollars, the more advantageous it is to transfer your credit card points to your airline miles account, and buy the ticket there. For example, that one-way United ticket to Europe we talked about earlier? Yeah, say it's on sale for $600. You can buy it outright through Chase for 40,000 points, or you can transfer 30,000 Chase points to United and then be done with it.

Overall, the more expensive the ticket, the more sense it makes to transfer your points to the airline and book through them. The pattern tends to support this approach, in particular, when traveling internationally since those tickets are almost always more expensive.

Fine Print and Gotchas

There are some caveats to keep in mind as you do this.

Always game out both options to see which tickets are available before you transfer points or commit to a flight. Some flights may not be avaiable using both options. Some more popular flights may not have open seats to book with miles, or they may charge a miles premium.
Remember that not all credit cards transfer to all airlines, so that a card partners with your airline of choice might make that a better choice for you. As of this writing, none of the big three cards were transfer partners to American Airlines.
Trips when you use your points like cash to book through your credit card are counted by airlines into frequent flier earnings and status. Trips you take by redeeming miles direct with the airline do not.
Most transfer partners work at a 1000:1000 ratio, meaning that the points transfer evenly from card to airline. Some, however, have less favorable ratios... so be sure you're getting the best deal.
Transferring points to airlines in this fashion will tend to be more lucrative on airlines that use a more traditional scheme for mile redemptions. Learn more about this in my previous post "Demystifying reward travel for "Traditional Miles" American, Delta, United, and Alaska Airlines".
Conversely, it makes much less sense to transfer to airlines like jetBlue or Southwest because they use a points for cash approach themselves, so you're essentially just moving money around with no added value. The exception is if the airline is offering a bonus to, say, transfer 1000 credit card points for 1200 airline points. Again, do your homework on the real value in each specific case. Read my previous post "Southwest's and jetBlue's reward programs offer simple ways to quickly earn free travel" for more info.
Beware the "fuel surcharge". British Airways is notorious for for charging you miles (they call them "Avios"), and then charging you hundreds of dollars of fuel surcharges on top. Overall it makes redeeming miles direct with airlines that do that a pretty bad deal.

...You must decide for yourself if you'd rather spend, say, 6,667 Chase points on a $100 ticket when you know you could transfer 30,000 to, say, United and fly one way to Europe. That really boils down to what you value and where you fly most often.

Happy travels!