Hello. I am an atheist; or "agnostic atheist", to be more precise for those of you who like to quibble about definitions.
First, I'd like to apologize in advance, as I am trying to write and publish this in the limited time I have remaining during my lunch break. I'm likely to make typos and/or fail to edit sufficiently, but at least deadlines focus one's thoughts.
Anyway, I'd like to explain how I, as a non-believer, think of the "afterlife". I am not claiming this is a new or original idea by any stretch of the imagination, but it's become more clear to me now, having been through the deaths of a few people close to me in the last several months; and it's really rather simple... One definition I hear bandied about frequently for a "soul" is that it's that part of a person that remains when their mortal bodies cease to function - which is not to say their corpse, but something more amorphous and immaterial - the "essence of their personality" so to speak. From there it's just a question of trying to understand what that might be exactly, in a real-world sense.
For me, the answer is this: our "soul" is that which we impart to others and to the world around us. It is the various kindnesses we have done for others in ways that have changed lives or the way other people think - not just about us, but in ways that may have impacted their personalities or world-views. We are, in one sense, the end product of those who've raised us and affected us. These teachers of ours who have passed (may they rest in peace) are still with us in this sense, because they are a part of us now and everyone else they touched over the course of their mortal existences.
On the other side of the coin, it's not all butterflies and rainbows. Rather than help, some people hinder and leave scars and various wounds. This is unfortunate, of course, but I'm afraid it's unavoidable.
In either case, my personal definition of an afterlife is simply that I "exist" as an echo down through time and the generations, for better or worse. I do not think there's sufficient evidence for me to believe that "I" will have any awareness of my post-mortem "existence", but it will be there none the less. An "afterlife" is, in other words, one's legacy (plus the legacies of those who are impacted by what I've left behind). For me, I hope to make it a positive one - one that brings happiness, joy, and fulfillment to those carrying my "echo" (that would be "heaven"), as opposed to worry, pain, uncertainty, suffering, and confusion (hell).
Thank you. Please be kind (to others, as well as to me in the comments should you choose to leave one).