Being a 23 year old entrepreneur, I am slightly biased when I make the statement that the best time in life to start a business is before your 25th birthday. That said, I still think I make a great case for starting a business sooner than later (please note, I understand that the statements I make don't apply to EVERYONE, however I feel they apply to the overwhelming majority of people).
By starting a business when you are young, you avoid procrastinating
Ever hear someone say "I'd love to start my own business someday but...."? Those are the people who ultimately won't start their own business. They are waiting for the "perfect" time to do it. The moment when all the planets align. It will never happen. You can always make up an excuse. When you are young, it's college loans that need to be paid back or a lack of experience. As you age, it becomes your marriage, kids, house, etc. There will always be a reason to procrastinate. By starting young you avoid becoming a habitual procrastinator.
When you are young, you have less responsibility
If you think it's hard when you are twenty, it will be ten times as hard when you are forty. By then, you will likely be married with kids, a house, a nice car, and a lifestyle that you are pretty comfortable with. Think about how difficult it would be to try to convince your significant other that you are going to take on something that will occupy nearly all of your free time and might not reap any benefit for a few years. It's tough convincing yourself to make the plunge into entrepreneurship, it's nearly impossible to convince your family to dive into uncertainty with you.
When you are young, you are used to a hectic schedule
College age students are used to all nighters. They are used to getting emails from a partner at midnight about something due at 8 am the following day, telling them that they did something wrong and need to finish it up by the morning. Somehow, they find a way to meet the deadline.
This is very conducive to the entrepreneurial environment. Your day does not begin at 8:30 and end at 5:00. It begins when it needs to and it ends as soon as the work for the day is done. Somedays, everything goes smooth and you can work your 8:30-5. Other days you work until after midnight. It is all dependent on things that are somewhat out of your control. No matter how good you plan, you can't plan for everything - a shipping mistake by UPS, two feet of snow hitting the ground out of nowhere causing you to be an hour late for an appointment, or your website crashing - unexpected things will always go wrong. Students are used to this. People who have worked the 8:30-5 for twenty years tend to check out mentally at 5 every day. You can't do that as an entrepreneur.
Students are used to being without money
College kids get by on Raman noodles and cheap beer. They live with three other roommates and walk to class. Most of them have part time jobs for some spending money, but they could probably get by on what they earn over the summer or what Mom and Dad give them. Why is this important? Because as an entrepreneur you are probably not going to see any income for at least six months (it depends on the business of course) and maybe for as long as a few years. For the college entrepreneur, not a big deal. For the guy who was used to pulling in $100,000 a year, it's a big deal.
When I started SportsLizard.com a few years back, I was able to do it with about $3,000 of savings from an internship I had the previous summer. I didn't make any money for a while, but it didn't really matter. You know why? I was living in my parents basement. When your cost of living is $0 it takes away a lot of the stress.
Young people are better learners
Everyone is good at what they do. Computer programmers are good at computer programming. Artists are good at art. Athletes are good at being athletes. What are students good at? Learning. That's right, students are good at learning. It is, after all, what they do for a living. They are used to seeing something completely foreign to them and mastering it in a short amount of time. Professionals tend to become more and more focused as they get older. While they are really good at one or two things, they haven't learned something completely new in a long time. The workforce has a way of unintentionally limiting you by narrowing your skillset. This works against them as an entrepreneur.
As an entrepreneur, you will be low on resources. Don't count on having a secretary to do the things that you don't know how to do. You can't afford a secretary! While your main focus is living your passion, something you are inherently good at, you will have to do quite a bit of learning to be an entrepreneur.
For example, it may appear that I run a website and the only real thing I needed to learn was in regards to web programming. I don't think many people would consider the amount of time I put in learning how to run my business. I had to research trademarks before choosing a name (you wouldn't want to pick a name that someone else already has), I had to figure out how to register my business with my county, and I had to learn basic tax laws to make sure I correctly report all of my income. I also handle all of the PR and marketing for my business, not to mention customer service and budgeting. There is a lot to running a business, even what appears to be a simple one that I run from my house (imagine if I had employees, rented office space, etc). A business owner is going to have to learn a lot of things quickly and be able to execute what they just learned with confidence. Young adults are better at this than any other age group.
Young adults are used to change
One of the reasons you will need to learn so much is that you will need to change often to compete as a startup business. College kids are used to moving every few months, changing classes every semester, and having different summer jobs each year. This comfort with change will make it easier to make the necessary changes for the good of the business. Someone who has lived in the same house and worked the same job for fifteen years will naturally be more resistant to change.
The other side of the argument
Obviously the downside that everyone will point to is the lack of business experience that young people have. If you can't tell already, I view this as an advantage. I feel that industry will focus you on what you're good at. For years people will tell you how good you are at what you do and encourage you to grow in that area. Ten years later you are a master of that trade but can't even conceive of switching professions. At that point, becoming an entrepreneur is a lot harder.