The importance of volunteer experience for the university application process varies. First, it will depend on the specific university you are going for. Public universities want to see groups and clubs you were in, events you helped plan, school fundraisers, etc. General things you do in High School. Basically they want to see that you occasionally did something you did not have to do that either helped others, broadened your social world, or both. Mandatory things don't usually count for much.
Private universities may want to see those things, but they are looking for more. They want to see that you went out of your way to to help your community, do things that weren't always pleasant, etc. etc. Chess club might cut it for your local university, but if you're going for bigger and better, you'll need more. Private universities know you have good grades, they know you most likely did well with social groups and events in high school. You have to be willing to go out of your way to make things jump out at them as they read. They see tons of applications each semester, you want something on there that will make someone go "how interesting".
Just a simple example: Public, volunteer Humane Society dog walker. Private, minimum of six months with fixed hours and a reference letter from your manager / instructor working as a veterinarian's assistant*.
Spending time volunteering means working without pay and often doing work that you may not enjoy, at least at first. What it needs toshow to a university, regardless of whether it's your little local one or Harvard, is that you were responsible, showed respect to those you worked with, always did what you were told, and showed a willingness to be open and learn new things. Because this is all the characteristics they want in their students. You ask "how important is it", and my answer is that it's not whether it is important or not, it's the quality of the effort. Helping your school raise money could be just as important is a volunteer job. They want to know that you worked as hard as possible and were kind and respectful to everyone around you. I wouldn't do anything with hope for putting it on a college application without making sure someone sees what you do and can provide a long, descriptive letter of reference for you. Otherwise, it only matters if the person reading it cares, and that varies from person to person.
*Humane Societies often looking for volunteers age 16+ to help them while giving medical care to the animals. This is someone who cleans up before and after medical procedures, watches animals after surgery, and may learn how to do basic medical tasks. This is not to be confused with a real vet tech, who has a degree and experience.