In order to visualize the 3-D helix structure, think of each of the components in physical terms... 

The phosphate is a compact inorganic salt (NaPO₄) - think of it as resembling a ball of charge...

The sugar can be visualized as a pretty tight hydrophilic knot of four C and one O.
(The ring itself has a particular pucker)

The Nitrogen bases are basically flat plates - aromatic 5 & 6 C rings.
Each Carbon in the ring shares two e^- with two adjacent C.
Each shares another two with an atom outside of the plane of the ring.
The final pair of e^- is shared with one of the two adjacent C in a double -bond.
This pair of electrons shifts between the two adjacent C, resulting in a delocalized electron pair
The delocalized e^- cloud occupies space above and below the plane of the ring (the p-cloud).

The overall picture is one where
the sugar-phosphate backbone wind around the outside of the helix
and
the Nitrogen bases stack over-top of one another down the center of the helix.
The stacking of the Nitrogen bases allows the p-clouds of adjacent bases to interact and extends the region of e^- delocalization. In addition, the stacking of bases excludes water from the center of the helix - further adding to the stability of the whole structure (hydrophobic interactions).

The diagram below illustrates the basic dimentions and arrangements of the B form of the double helix.

The B form of the DNA double helix is not the only conformation DNA can assume.

When dehydrated, DNA takes on the right-handed A-form conformation in which the bases are tilted with respect to the long axis of the helix. The A-form is believed to be the conformation of DNA-RNA hybrids, and may represent the conformation of the template when being transcribed.

Additional conformations of the double helix have been described for specific sequences.
A left-handed Z-form has been described for sequences of very high GC content.
Regions of high AT content embedded in random sequence have been proposed to introduce a bend in the helix which can affect fragment migration through an agarose gel.

Three forces are responsible for holding the double-helix together.

First, the H-bonds between the complementary bases hold the two strands together.
Note that the G:C content of the sequence will influence the overall stability of the helix
as G:C pairs share 3 H-bonds while A:T pairs only share two.

Hydrophobic interactions also play a large part in stabilizing the helix.
Since the aromatic rings N-bases are hydrophobic, these interactions keep the bases confined to the center of the helix where they are protected from interacting from the polar solvent.

Finally, the stacking of N-bases down the center of the helix adds additional stability.
The stacking energy is a combination of Van der Waals forces between the p-electron clouds of the aromatic rings (a hydrophobic interaction) and the interaction of the dipole groups.

In general, the forces holding the helix together are not particularly sequence specific.
The stacking energy is an exception to this rule. Because of the way the bases are oriented down the center of the helix, the amount of p-overlap and dipole proximity will depend on the actual sequence.
An example of this sequence dependance is shown below.