Simply put, introverts face the daily challenge of feeling like square pegs being shoved into round holes. Introverts are a minority in this extroverted world, much in the way that left-handed people find that a lot of things are designed with right-handed people in mind. Let's take a look at the challenges of being an introvert.
Let's look at its opposite, extroversion, first. Extroversion is a personality attribute whereby an individual thrives in the presence of other people and actively seeks out other people to socialize with. They feel energized by the presence of other people. Extroverts are often bored and unhappy when they are alone. They literally need other people in order to survive and thrive.
Introverts, by contrast, need their own space and time to themselves. Being around other people can drain the introvert's energy, leaving them tired and irritable. They need time with their own thoughts and regain energy in this way. Introverts need to feel that other people, especially extroverts, understand and accept these needs. However, this can be very hard to acheive when there is a lot of social pressure to attend social gatherings and spend time with other people.
By nature, the introvert is comfortable being alone. Introverts need time to think their own thoughts and dream their own dreams without interruption or criticism. This can be harder to do than we might think for a class of people often thought of as loners. This is particularly true for those introverts who have family obligations. For example, young children need, and demand, quality time and attention. During the course of a day these frequent demands can become almost unbearable to an introverted parent. Despite the love an introverted parent will feel for their child, the continual drain from the child's need for attention and contact can seem like an annoyance. Unfortunately, the child doesn't understand: he or she only knows that they have a need to be fulfilled and their parent isn't doing their "duty".
The essential challenge facing the introvert is that you will continually have to fight your natural inclination toward solitude in order to function in society or within a family. The introvert must therefore find ways to "recharge" and get the quality alone time that they need to stay healthy and happy. It's the way we are wired.
One final note: an excellent article that really helps to explain the introvert experience is Caring for Your Introvert by Jonathan Rauch.