A lower limit on the age of the universe is a value representing the minimum age for the universe. Lower bounds usually take the form of "Because we observe X, the universe must be at least Y years old". Astronomers, astrophysicists, and cosmologists have several tools to estimate the age of the universe.
 Globular clusters
Logically, stars can't be any older than the universe that they occupy. Finding the age of the oldest stars or groupings of stars is thus one way to determine a minimum age for the universe. Old stars can be found in globular clusters—groupings of a few hundred thousand to one million stars or so.
Fortunately, the lifetime of a star is related to its mass. Massive stars burn out quickly; those that are 10 times more massive than the sun burn out in 20 million years. Stars half as massive as the sun can last 20 billion years.
If we assume that the stars in a given globular cluster formed at roughly the same time, and that there was a reasonable distribution of stellar masses in the cluster initially (reasonable assumptions, based on what is known about star formation), then we can get an estimate of its age by examining the stars it contains. We can estimate star masses by evaluating their luminosities and distances from earth. If we observe no stars more massive than ten solar masses, indicating that all of those big stars burned all of their fuel and became something else, then the cluster must be at least 20 million years old. If we saw clusters containing only stars half as massive as the sun or less, then the cluster would have to be at least 20 billion years old.
The oldest observable globular clusters contain stars of 0.7 solar masses or less, which places them between 11 and 18 billion years old. Uncertainties in distances prevent greater precision. Taking the lower estimate of 11 billion years for the age of these clusters, the universe must be even older than that, since it took time for the cluster to form in the first place. Taking the cluster formation time into account pushes the lower limit for the age of the universe based on the age of globular clusters to 12 or 13 billion years. Even if we hedge on that estimate, we can be confident that according to scientific evidence, the universe has existed for more than 10 billion years.
According to this estimate, the scientific evidence differs from the YEC assertion of the universe's age by a factor of about 1 million.
 Light speed and the visible universe
Assuming that the speed of light has always been constant, if you can calculate how many light years away something is, it has taken light that number of years to reach the observer (us).
The furthest distance away that deep space telescopes can see is somewhere in the region of 13.7 billion years (approximately), this implies that light has taken around 13.7 billion years to travel across the universe to reach us. Thus, one concludes that the universe is at least 13.7 billion years old.
Unless the speed of light isn't constant... 
 See also
- Last Thursdayism
- Evidence against a recent creation
- Branches of science you have to ignore to believe in Young Earth Creationism