What is William Cullen Bryant's poem "Thanatopsis" talking about?

This is actually a pretty famous poem by William Bryant, but famous doesn't mean the same as "easy to understand."  I think I can point you in the right direction on the subject, though.  I won't go line-by-line like I usually do when it comes to analysizing a poem (Thanatopsis is just too long to do that with) but I still think I can give you an overview of the poem's "meaning."

The theme of the poem is about death, and the author's opinion is that one should not fear it.  That seems contradictory to the first section of the poem...most of the images used paint death as being pretty unpleasant.  Take a look at these quotes to see what I mean:

"Of the last bitter hour come like a blightOver thy spirit, and sad imagesOf the stern agony, and shroud, and pall,And breathless darkness,"
"in the cold ground,Where thy pale form was laid with many tears,"
"The oakShall send his roots abroad, and pierce thy mould."
So from these quotes it would appear that Bryant takes a negative view about death (I mean, just take a look about the one that says trees are going to poke their roots into your corpse for nourishment!)  But this is not the point of the poem, as evidenced by the second section...
"Yet not to thine eternal resting-placeShalt thou retire alone,"
"All that treadThe globe are but a handful to the tribesThat slumber in its bosom"
"As the long trainOf ages glides away, the sons of men,The youth in life's green spring, and he who goesIn the full strength of years, matron and maid,The speechless babe, and the gray-headed man--Shall one by one be gathered to thy sideBy those who in their turn shall follow them."
This section of the poem says, basically, that although you are going to die you will do so in good company.  Everyone has to die, and when you do it you will be going to the resting place of geniuses, Kings, babies, and other good folks.  Even those who laugh at you will, in time, die too.  In fact, Bryant says, the number of people on Earth who are alive is nothing compared to the number who have died already...death is just the act of joining this giant crowd.
The last part of the poem sums up this feeling.  Having an understanding that death is just joining this giant mass that has gone before you, you should go to your death bed : "Like one who wraps the drapery of his couchAbout him, and lies down to pleasant dreams." In other words, do not fear death.  Live your life in a good way and then go to your death knowing that you are strolling down a path that has been well worn before you.  That thought is supposed to provide you with comfort.
Hope this helped!