One of the best ways I've heard for explaining meditation is to think of your mind as a big lake. It's something I've read in Deepak Chopra but also various other places.
If you think of your mind as a big lake, your thoughts are creating ripples on the surface like stones being thrown in, and some of them even create ripples deeper down. Meditation helps you to calm the surface of the water by having fewer thoughts. The idea is that then, you will be better able to notice your individual thoughts and concentrate on what needs to be done - rather than trying to pick out individual ripples from a surface that looks like 500 stones have just been thrown in.
Andy Puddicombe also talks about how our minds are like the clear blue sky. There may well be a few clouds in the sky - or even so many clouds the sky appears to be grey - but if we can get above the cloud of our thoughts, that calmness, that clear blue sky - is always there, just behind what we can see. Anyone who's been on an aeroplane can tell you this.
So although you might feel like you're wasting your time, spending half an hour sitting in silence and trying not to keep thinking about the washing up - but really the very act of sitting quietly is helping.
Sometimes, at the end of a stressful day, meditation feels to me like someone is reaching into the top of my head and pulling out all the debris that has collected there, and taking it away. After meditating I am able to think clearly and am less easily distracted.
A 2012 study at Massachusets General Hospital found that 8 weeks of daily meditation can have lasting effects on the brain and the way it works - even when you are not meditating. I have certainly found a marked improvement in my general mood, and also my ability to handle stress, since I've been meditating. It's slowly changing my whole life - from big things like my outlook on life, to smaller things like "hmm, I don't think I'm going to drink coffee any more..."
In this world, in Western culture in particular, we seem to spend a lot of time, effort and money in attempt to distract ourselves from our thoughts, to disconnect from what is going on in our heads. We watch crappy TV that half the time doesn't even interest us; we listen to the radio; we play video games; we get drunk or take drugs. I really think the world as a whole would improve greatly if we could all just learn to sit with our thoughts instead of trying to hide or get away from them. Meditation is a part of that; it helps to stop your mind from racing so fast that you can't keep up with the thoughts. It helps you to slow down and look at things logically. It calms you down.
The meditation I do is primordial sound meditation; I did an online course through The Sleep Guru, who presents it on behalf of Deepak Chopra.
If you would rather not go down the primordial sound, spiritual path, Headspace has some great free guided meditations for mindfulness, and also has a free app for your phone.
If you decide to try meditation, do please let me know how you get on; I'd love to hear about it.