Synergies are skills that boost other skills or that are boosted by other skills. Synergies have always been part of the skill trees of Diablo II, but their properties become more tightly integrated and explicitly documented in the v1.10 patch. In v1.10 and later, synergy bonuses come only from actual skill points in a skill; +skill bonuses from equipment do not count towards the synergy bonuses.
How Synergies Work
Synergy relationships can be complicated and work in both directions.
For example, each point in the Paladin combat skill Sacrifice adds 12% damage to Zeal. Also, each point in Redemption adds +15% damage to Sacrifice, and each point in Fanaticism adds +5% Damage to Sacrifice. A character who wished to use Sacrifice as their main skill would therefore want to max it out, and also max out Fanaticism and Redemption, and at that point they might as well put a few points into Zeal, since the 20 points they'd sunk into Sacrifice would make each point in Zeal quite effective. This character would do very high damage with Sacrifice, but since they'd spent 60 skill points to develop that ability, they have very little left over for other skills.
Synergies should be looked over carefully before allotting your skill points -- in this instance points in Zeal only add 6% damage (they add 10% AR also), while points in Sacrifice add 12% damage to Zeal. So if your goal is to do high damage with Zeal, you should put in enough points to get 5 hits, and then work on maxing out Sacrifice, even if you never intend to use Sacrifice itself.
Many other skills work this way, encouraging or almost forcing a player to max out 2 or 3 or 4 skills in order to make just one of those skills work especially well. This feature of the game has proven somewhat controversial, since it basically requires players (who wish to build a powerful character) to invest all of their points into just a few skills, yielding rather cookie cutter results.
Synergy Pros and Cons
One the one hand, synergies they allow characters to specialize their skills and greatly improve the damage output of one skill or type of skill. On the other hand, this causes (forces?) characters to become very one-dimensional, with all of their skill points devoted to a single skill or skill tree.
For instance, most Sorceresses were bi or tri-elemental prior to v1.10, with a major attack spell each from fire, cold, and lightning, or at least from two of the trees, to better deal with monsters that had high resistances or elemental immunities. This allowed much wider variety in character design and play style.
In v1.10+ tri-elemental sorceresses are almost unheard of, since that build means they do considerably less damage with their biggest skills than a synergy-specialized Sorceress will with her one skill/tree. And since the hit points and resistances of monsters were boosted in v1.10, tri-elemental or bi-elemental sorcs can't kill very effectively anymore, while the new style, one-tree, synergized sorcs are devastating to any monster not immune to their element, and helpless against any monster that is.
In effect, the synergies require specialized characters to play in mixed parties, while making other, more varied character builds ineffective. Player opinions differ widely as to whether this is a mark of progress, or a design failure.