As someone who has suffered from chronic depression since as young as six years old, and first started having "those" thoughts at age eight:
Someone early on in this thread spoke the truth, that people in a serious state of depression honestly believe others would be better off without them. Complicate matters with my physical disability. I cannot hold a job, and have been told twice by Vocational Rehabilitation that I am unemployable. I'm so disabled I can't even work through programs designed to find jobs for the disabled. So when I see my husband struggling to make ends meet, and I can't help him pay the bills, how useless do you think that makes me feel? It's worse when he comes home from work and then has to do a chore that was too much for me, and worst of all when some medical need of mine throws his carefully calculated budget out of whack. Are there times I think he'd be better off without this burden, and I wonder what in the world he sees in me? You bet there are. And when I am at my gloomiest, I can't even imagine that anyone would grieve for me. I actually picture them dancing and singing, "Ding Dong, the Witch is Dead."
Depression is not just feeling sorry for yourself because you didn't get asked to the prom. In fact, that's kind of an insult. That's comparing acne to advanced skin cancer. True depression is a malfunction of the brain. When the brain isn't working properly, thoughts won't make sense except to the person who is thinking them. But they are honestly believed.
One illustration I use is the concept of visual hallucinations. For example, let's say I am hallucinating and I see a snake on the floor in front of me. No way will you ever be able to convince me that there is no snake. What do you mean, there is no snake? I see a snake plain as day, doggone it, and you can't tell me I don't see it. What are you trying to do, mess with me or something? If you don't see something that is obviously right in front of you, then you must be the one who's nuts. Only when the brain chemicals quit misfiring, by whatever means it takes to accomplish that, will I stop seeing the snake. Depressive thoughts work the same way. Even though they are not accurate, they are just as real to the person with depression as that snake is to the person who is hallucinating.