Forgotten Realms hews reasonably close to the generic concept of D&D worlds. Because of this anything that you may find in an official 2nd, 3rd, or 4th D&D book is probably around somewhere (or was.) The primary difference is that Forgotten Realms has been put through 30 years of being used by roleplayers, authors, and computer game writers.
So what is Forgotten Realms? Forgotten Realms refers to the world, and associated extra-dimensional planes of existence linked to that world, known as Toril. Currently the planar situation can be explained thusly:
The prime-material plane consists of the world as humans and the other common races (elves, halflings, gnomes, dwarves, dragonborn, eladrin, goliaths, etc. etc. etc.) know it. This world has magic in it, both arcane and divine, but is principally a natural world where the weather and the seasons go through the standard cycles. Climate variety is much like Earth, as are many of the cultures abounding in the world (being drawn from a variety of locales and times in our past).
The prime-plane has two echoes that are almost distinct planes of existence in and of themselves. The bright echo is the Feywild, a world of beauty and nature. Everything in the Feywild is MORE itself, more perfect. Food always tastes incredible, the vistas are all stunning, and magic is potent and easy. The dark side is that the evil of the Feywild is more evil, villains more foul, and plots more sinister. Geography tends to match that of the real world, though distances can be deeply distorted, and no artificial creations (such as towns) are reflected in the echo. There are, however, cities that exist solely in the Feywild, places of magic and grandeur unknown to the prime-plane. The dark echo is the Shadowfell, a plane that is a grim reflection of all that is. Great cities may appear as ruins if at all, proud mountain peaks worn away. Shadowfell is a place of death and darkness, yet like the Feywild is much of a world unto itself. There is life here and indeed there are cities and cultures. All are grayed by the oppressive nature of their surroundings, but the Shadowfell is no more a place of evil than is the Feywild a place of good.
Above and below, in the literary sense, lie the two principal planes associated with the prime-material one (other than the twin echoes of the Feywild and the Shadowfell). The Astral Sea is home and birthplace to the gods and is a plane dominated by pure thought and subjugated through strength of will. Its opposite is the Elemental Chaos, a realm of constant churn and primal chaos. The prime material plane was fashioned long ago by elemental Titans as a sort of hobby, but it was the gods that gave it permanence and a status beyond the transitory nature of the elemental chaos. It is this unique combination of the raw stuff of creation and the power of the astral sea that sets apart the prime material plane and the creatures who live upon it. The twin nature of body and soul reflect this dual heritage.
Finally, because all of this isn’t complicated enough, there is a twin world to Toril called Abeir. This is a result of the struggle between the Gods and the Titans when the world was young, and so the world was split with the Gods dominating Toril and the Titans Abeir. This has consequences later on, particularly due to the Spellplague. Usually the two worlds are generally inaccessible from each other.
[Stop, take a breath. Your characters don’t know most of this anyway. Alright, feeling better? Let’s continue.]
About one-hundred years ago a calamity overtook the twin worlds of Abeir-Toril. Mystra, goddess of magic and the protector of the Weave (a sort of magical web that mediated the interaction of mortals with magic) was assassinated by a cruel and insane rival. Without Mystra’s harmonizing influence the Weave began to unwind, wreaking incalculable havoc across the worlds. This was the origin and prime disaster of the Spellplague. Thousands of wizards, mages, and sorcerers went insane, died instantly, or were cut off forever from their life’s endeavor. The many cultures and institutions that depended on magic were affected in unequal measure but were just as often shattered as miraculously saved. Divine spellcasters, those that received their power from the Gods, were cut off from their patrons due to the turmoil of various planes shifting about Abeir-Toril (including the formation of the Shadowfell as the Plane of Shadows melded with something… else, perhaps the Plane of Negative Energy… don’t ask). Cut off from their patrons’ power, organized healing, through the aegis of priests and other divinly associated spellcasters, collapsed throughout the realms for years, throwing the faithful into turmoil and prompting many prophecies of armageddon. Whole countries were torn apart by wild magics, while others were replaced, swapped for locales in Abeir, or dredged up from the depths of time. What had been an increasingly peaceful and civilized world was thrown back into turmoil, such that the modern world very much resembles a darkness lit by intermittent points of light, vestiges of old civilizations and embryos of new powers.
This is the world in which your characters live. A world that is sometimes hostile, sometimes wild, but also a world with unlimited promise for the future as it tries to rebuild.