NOTE: This section is pure speculation based on clues provided in the TV Series and in the Movies except where references are cited.
The official definition from the producers is "An Immortal can only be killed by beheading. When an Immortal is killed, the Immortal who killed him receives his power and his essence and his knowledge through a mystical process known as "The Quickening." The power of the Quickening is the equivalent to a major electrical storm hitting -- windows explode, lights short circuit, it is almost as if the victorious Immortal is in the center of a lightning storm." Traditionally 'quickening' is a term used to describe the first discernible movement of a baby in the womb, i.e. when the fetus first shows signs of life. It has been theorized that Greg Widen chose this term do describe the peculiar life force of Immortals. As Adrian Paul describes it, "The Quickening is the receiving of all the power and knowledge another immortal has obtained throughout his/her life. It is like the receiving of a sacrament or a massive orgasm. That's my motivation."
Ramirez tells Connor "You are only safe on holy ground. No one will violate this. It's tradition." Their Code will not allow them to fight in sacred locations. The consensus of all the discussions is that Holy Ground is any place held sacred by any religion.
Ramirez told Connor "You must learn to conceal your special gift, and harness your power (pause) until the time of the Gathering... When only a few of us are left, we will feel an irresistible pull towards a far away land, to fight for the Prize... We must fight until only one remains..." During the movie, we assume that the Gathering lasted only a few weeks, at most a few months. But, in the TV series premiere, we are told that the Gathering has begun and, in the second season premiere, we are told that Connor killed the Kurgan eight years ago. So presumably the Gathering will last several years, or at least as long as the TV series :-).
Immortals differ from mortals as follows: they (male and female) are sterile from birth, they heal quickly from almost any wound, they don't age, they can only be killed by decapitation, and they can sense each other's presense by an internal Buzz they feel when they come in close proximity to each other (a distance ranging from ~20-~100 feet).
An Immortal will 'die' from anything that will kill a mortal. But they regenerate and recover. The more extensive the injury, the longer it takes to regenerate. Scars received before the first 'death' remain but scars from almost-decapitating wounds don't disappear (Kalas in "Song of the Executioner" and the Kurgan in the original movie). We found out in the 2nd season episode 'Unholy Alliance Part I' than an Immortal doesn't seem to be able to regenerate a lost limb (Xavier St.Cloud is sporting a hook). Fans have speculated that the hand may have reattached if it could have been held against the stump for a period of time. And others have speculated that maybe it takes longer than a year for a limb to regenerate (There was about a year between "For Tomorrow We Die" and "Unholy Alliance Part I").
Immortals (male and female) are sterile from birth. An Immortal carries the "seed" of his Immortality within him in life. If he is "killed," he becomes Immortal and never grows any older and heals quickly.
They can sense each other's presence. The producers call it "The Buzz."
Immortals appear to have all the characteristic of mortals when they are born except they are sterile and have a very faint Buzz. They cannot sense other Immortals with the Buzz. Their healing abilities and other Immortal traits remain latent until they receive a mortal wound.
One major difference in Immortals physiology between the movies and the TV series is whether or not they 'die' when they received a normally mortal wound. In the first movie Connor seems to get near death ("You, talking and breathing, and last night all but a corpse" -Dougal MacLeod in the tavern after the battle with the Frasers and the Kurgan) but don't actually die. Connor MacLeod even appears to breathe underwater. Whereas in the TV series, Immortals appear to actually die and then recover/reawaken. TV Immortals have been pronounced dead by modern medical professionals; Felicia Martins was transported all the way to the City Morgue before she recovered in the first season episode 'Free Fall'. And Duncan was buried and then unearthed by Lucas Desiree in a flashback in the first season episode 'Innocent Man'. Even Richie makes a trip to the morgue in the third season. But they all recover quickly (rate seems to be dependant on the severity of the damage that caused the 'death') and their wounds heal.
Also in the TV series we've seen two instances in which a wound on an Immortal has healed with sparks and little lightning effects (Duncan in 'Deadly Medicine' and 'Counterfeit Part II').
In the TV series if a mortal decapitates an Immortal nothing happens, no Quickening. To quote Hugh Fitzcairn in "The Hunters" (first season finale): "This...this is madness! There are no Immortals here. All I have will be wasted! What do you want?!?! There will be no Quickening! What do you want? You are mad!"
A sword is one of the best way to ensure that when an Immortal takes the head of another he will be close enough to receive the Quickening. As with any weapon, to use it effectively you must practice to maintain your skill. So Immortals probably prefer the weapon that they are the most familiar with. Considering the age of most Immortals, it stands to reason that the sword is usually the weapon of choice.
In addition the relationship between an Immortal and his sword (or other prefered weapon) is almost a mystical one, considering the way they almost always them on hand and their weapons also seem to be connected to the Quickening process. For example, the sparks from the swords when they contact an opponent weapon or other metal object during a battle with another Immortal and, at the end of 'Innocent Man', a small flash of lightning appeared on Lucas Desiree's sword from atop his grave where Duncan had planted it, indicating some residual force left over.