An Insert

A close-up shot of hands, a document, a murder weapon, and so on. Cut into a scene to appear part of the continuity, but probably shot separately. The hands often don't belong to the actor in the scene.

If the hands are effecting a murder, they are Hands of Death. Especially if clad in black gloves. When it's the director doing this, it's an Insert Cameo.

Examples of An Insert include:

Film

In Titanic, the hands drawing the picture of Rose belong to James Cameron.
Phantom of the Opera: The creepy, black-leather-glove-wearing hand locking Christine's dressing room belongs to the Phantom, naturally.
The character introduction in Monty Python and the Holy Grail is done by means of an expository book, the pages of which are turned by a cast member's wife. At the end of the shot, the hand is snatched away by a gorilla hand.
- All further shots of the book are then turned by the gorilla hand.
The Sting. When Henry Gondorff (Paul Newman) is doing card tricks before the poker game, the hands doing the tricks are actually those of John Scarne, famous magician and expert on gambling and card manipulation.
Similarly, in The Flim-Flam Man magician Jay Ose's hands did the card flourishes for George C. Scott
Done in Labyrinth whenever David Bowie manipulates his magical balls. Now get your mind out of the gutter. The hands shown are those of juggling artist Michael Moschen.
The DVD Commentary to Love Actually points out a "very fluid" one, of the videotape being put in the VCR, in the scene where Juliet goes to Mark's house.
Quite common amongst films portraying a character playing a musical instrument. Ralph Macchio played none of the guitar parts in Crossroads, those were played by Ry Cooder.
The Godfather: When Michael opens the newspaper to read about his father's shooting, his hands in the insert are played by George Lucas.

Live Action TV

The reality show The Amazing Race features close-ups of hands holding the various clues/tasks the racers receive while host Phil Keoghan describes them in a voiceover. These are known as "Exposition Hands" to Television Without Pity members.
Knight Rider features several bizarre "Exposition Hands" sequences. At least once, for no clear reason, rather than doing an insert of an indicator on the dashboard, a close up of the indicator is shown on the dashboard's monitor. Several times, an insert of some feature being activated shows a hand pressing a button on the dashboard, even though there is no one in the car at the time. This latter is probably because the activation of the feature was associated with Stock Footage that didn't take into account the fact that KITT could drive himself when necessary.
- This was also used to allow functions and devices (such as the nearly-shoebox-sized analyzer and the later buttons for Super Pursuit Mode/Emergency Braking System) that would never fit in the actual dashboard. This has caused trouble for replica builders trying to be as accurate as possible.
An interesting example from Star Trek: The Original Series: actor James Doohan (Scotty) had seen service as a pilot in World War 2, where he had lost the middle finger of his right hand in combat. Keen eyed viewers will note that in close-up inserts of Scotty operating machinery, his hands have all their digits (because the close-ups were doubled by another actor). But in long shots, its more obvious that one of them is missing... this applies to all the movies featuring the original series cast as well.
The original Dragnet television series often had entire episodes composed almost entirely of inserts showing the recovery of evidence as Friday narrates, probably because the scripts were being adapted from the exposition-heavy Radio Drama. Stock Footage abounds, with the result that several episodes show the exact same footage of the same actors talking as Friday monotones, "Interviews with the friends and neighbors of the victim turned up no new leads."
The puppet-based action show Thunderbirds would often insert a live actor's hands when a puppet character would activate a switch or button.
- As a rather bizarre Shout-Out to this, the Live Action movie used puppet hands in one closeup.
Mercilessly spoofed in the Australian series Russell Coight's All Aussie Adventures, where a running gag involved the same insert of a handshake being played every time the title character met someone new in the bush. This is despite the new character being, variously, black (the handshake was between two white men), wearing gloves, sitting, or openly hostile to the main character (and thus not going to shake his hand).
The Sci-fi reality prank show Scare Tactics uses these more and more often as the show goes on. Considering the fact that the show is filmed by hidden cameras, you can bet that almost any embellishing shots (close-ups of equipment being operated, "special effects" sequences that the victim can't possibly even witness, etc.) were added in later. Well, either that or it's staged, but who knows...
A Lost example of a insert changing the plot: when Ben killed his father, it was originally intended that he would just sit there as the gas that killed DHARMA killed him too. They decided him having a active role in the murder would work better, so instead we get close ups of hands holding a gas canister...
In the TV Movie My Breast Meredith Baxter got a breast exam on-camera. The hands doing the exam were those of her doctor, not the actor playing the doctor.
An episode of The George Lopez Show had a hilariously blatant example, with the teenage niece displaying mad shuffling skillz at a poker game.
Mentioned in the DVD commentary of the 1st season of Stargate Atlantis, which explained that the same hand actor is used for all inserts, thus giving big fat fingers to otherwise delicately-handed actor David Hewlett.
On the Mystery Science Theater 3000 episode "Riding With Death," this device was used a handful of times so additional dialogue could be added that would tie together the completely unrelated plots of the two episodes smashed together to make the movie. Notably, a close-up of a speakerphone as we hear Ben Murphy say "Leonard, I hear you've grown quite the mustache while I was away."
Done particularly ineptly in an otherwise touching episode of Kamen Rider Den-O. The hands playing piano in close up clearly belong to someone a good forty to fifty pounds heavier than rail-thin lead Takeru Sato.