This question came from a former student asking if some forms of cardio are better than others. Now this question can be viewed from different perspectives. First, let's clarify that "cardio" (a term I dislike) means some sort of aerobic, endurance activity. Usually people think of this as steady state exercise. Steady state means that you maintain the same intensity throughout the exercise period.
My first instinct when asked this question or its variant of "What is the best cardio to do?" is to say that the best exercise is the one you are willing to do. I say that because, for most people, the primary goal of cardio is to increase heart health. From the heart's perspective, the mode or form of exercise makes no difference as long as it is continuous and somewhat repetitive and increases the demand on the heart. The heart just knows that it is being asked to perform "harder" and will adapt accordingly. So whether you are riding a bike, rowing on the erg, running or swimming you are placing greater demands on the heart (and the entire cardiovascular system). Also, there is no rule that says you need to do just one mode of exercise for general fitness. From a heart standpoint, if you ran one day, cycled another, rowed on yet another and then did a fourth day of the elliptical, you are doing just fine.
Are there differences in the modes of exercises? ABSOLUTELY! I would say the first thing is the preferences that you have. If you hate running, then find another activity! Some people have legitimate medical issues that eliminate certain modes of exercise. I know a man who broke his neck many years ago. The doctors warned him that if he crashed riding his bike and injured his neck in just the right (wrong?) way that he could be paralyzed. Some people have joint issues that make running painful, if not impossible.
People who are severely overweight might not be good candidates for running either at least not until they lose some weight. For these folks, exercise that does not increase demands on the joints might be preferable. Once the person loses some weight and can run, and still wants to run, then have the person start with a walking program to progress to a running program.
There are also practical considerations. Walking and running are relatively inexpensive sports. Invest in good walking/running shoes (please see a specialty store like Colorado Running Company in Colorado Springs or GoGoRunning in Rome, GA) and then get out! You can walk or run in almost any place with ease and the energy expenditure per minute is pretty strong. Walking and running has been shown to be beneficial in building or maintaining bone mineral density (strong bones).
Cycling requires, well, a bike. Yes you can start with that old bike your kid last used a few years back , or go to Wal-Mart and get a bike. However, you will probably "out grow" that bike and be looking for a good bike. You are going to invest a few hundred dollars (or even a few thousand) on a bike plus some equipment like cycling shoes, shorts, and a helmet (please always wear a helmet). You also are going to be limited in terms of weather in riding outside. (Ok, I was a wimpy cyclist and would not ride in rain or temperatures below 50 degrees F). If you travel a bike might not be the best choice. Cycling will not do much to help with bone mineral density and in extreme cases (think professional cyclists) might end up costing you bone--and not just from crashes.
Swimming requires a pool. Building your own pool will be pretty costly. Likely you will buy a membership to a facility with a pool. That pretty much limits your swimming that location (or maybe locations in a situation like a Y membership). Again, with little impact on bone mineral density.
Bottom line, as much as I love running, it is not for everyone. From a cardiovascular health standpoint just get out at moderate intensity (think a brisk walk) for 150 minutes per week to take good care of that ticker.