The Crossbow is a ranged weapon, and unlike the bows, it uses bolts. The crossbow unloads a bolt instantly, then automatically reloads post-shot without hindering it's user's movement speed. This 'cooldown' period, known as draw speed, can be brought down to an absolute minimum of 0.25 seconds through the redstone (haste) modifier and/or materials from other mods. Because there is no manual draw, the crossbow can automatically fire bolts at the interval of its reloading--just hold right-click! The crossbow only needs to have the bolts present in your inventory to work, as with vanilla bows. If the crossbow is not firing, make sure that the bolts are fully-made (they require fletching).
Because there is no manual draw, the crossbow compensates the lack of a critical strike (that a longbow or shortbow get from pulling the string all the way back) by instead replacing it with a reduced chance to critically strike.
The Crossbow is pretty expensive in materials - it needs a bowstring, a tough binding, a Crossbow Limb and a Crossbow Body. This weapon can only be made in a Tool Forge, and therefore is less available than the Shortbow.
The Crossbow consists of four different parts, and they all affect the bow differently. Please note that certain modpacks (FTB Infinity Evolved, for example) have implemented their own material re-balances, so the best materials for each part may vary from modpack to modpack. The material recommendations here only reflect the base statistics for Tinker's Construct and some addons.
Crossbow limb: The crossbow limb affects melee attack damage, draw speed, and arrow speed of the crossbow. Keep in mind that it is best not to use 'harder' materials in this part, like manyullyn, alumite or steel--this is because the reload time is increased with harder materials. The 'softest' materials have the fastest draw speed, ranging from wood to slime to even paper.
If Tinker's Construct is installed standalone, paper has the best draw speed (despite poor arrow speed) and steel the highest arrow speed (despite poor draw speed). Slime and blue slime offer a decent balance of both, plain slime erring toward better arrow speed and blue slime erring toward better draw speed.
With Minefactory Reloaded installed, the material with best draw speed is Pink Slime, which also has excellent arrow speed. With Thermal Foundation installed, "Shiny" has the highest arrow speed, but dismally slow draw speed.
If ExtraTiC and Thermal Foundation are installed, Signalum works even better--bringing draw speed instantly down to its absolute cap of 0.25 seconds, even despite a fiery bowstring draw speed penalty. Signalum also offers the very highest arrow speed, higher than both Shiny and Pink Slime.
Crossbow Body: This component only affects the durability of your crossbow, using the durability modifier that a material's tool rod or tough rod would impart to a tool. Thus, the best options for high durability are Manyullyn (or Pink Slime if it is available)--these both give 2.5x durability. Thaumium is a decent alternative part here, as it grants an extra modifier and 1.3x durability.
Tough Binding: Oddly, the tough binding has no effect on the crossbow's resulting durability, despite having an effect on other tools' durabilities. For this reason, the paper tough binding is often the part of choice.
Bowstring: see Bowstring.
Because it is a ranged weapon, the crossbow interacts with modifiers in non-typical ways.
Applying the quartz modifier will increase damage done by whacking a target with the crossbow itself, but this modifier has no effect on the crossbow's bolt shot. Adding lapis lazuli will result in more drops from mobs killed by either whacking the mobs with the crossbow itself or by killing mobs using the crossbow's bolts. Beheading, however, will only apply to using the crossbow as a melee weapon-- not to killing mobs with the loaded bolts.
Redstone decreases draw time, down to the cap of 0.25 seconds, at which point the crossbow will accept no more redstone. This makes redstone one of the most valuable crossbow modifiers, alongside lapis lazuli.
Knockback, fiery, and lifesteal are all noteworthy modifiers because, despite not applying to the bolts fired by the crossbow, they can be used to 'recover' in melee range by whacking mobs with the crossbow itself. Knockback puts distance between hostile mobs and the player, fiery ignites those mobs to deal damage over time, and lifesteal adds health back to the player.