The trace is a more sophisticated tracking charm than the Tracking Charm.
It can pinpoint the approximate location of an underage Wizard and, centered on that underage Wizard, detect magic within a certain range around him.
When an underage Wizard casts a spell in front of a Muggle, the Improper Use of Magic Office will be alerted and send a written warning directly to the individual. (Harry Potter's first warning, received without investigation, is a prime example.)
How is the trace created?
The Harry Potter books do not explain it in detail. [Most fanfictions assume it's on the wand, and many people mistakenly believe this is the original setting from the books, which is quite frustrating.]
The setting in this book is: it's created through contract magic.
The original text is: Notices were sent to every student, warning them not to use magic during the holidays.
Hogwarts students receive a notice from the Ministry of Magic during their summer holidays, prohibiting underage students from using magic outside of school. (This reminds me of similar summer notices I've signed before, and that's where the inspiration came from.)
Third-year students need a guardian's signature to go to Hogsmeade, so this signed notice isn't entirely unexpected.
This document contains contract magic, and the Ministry of Magic uses this contract magic. Of course, the magiccontract is not for prohibition, but for tracking and early warning, creating surveillance over underage Wizards, which is the tracking charm mentioned earlier.
When an underage Wizard uses magic in a Muggle-populated area, or in front of a Muggle, or if someone nearby uses magic, the Ministry of Magic's Improper Use of Magic Office will immediately receive a notification.
In the Harry Potter books, there is no indication that an underage Wizard's trace is related to his wand.
They only use the Prior Incantato charm to check the perpetrator's wand to determine what magic he used.
If the trace were related to the wand, I imagine one could simply switch wands to escape the Ministry of Magic's tracking.
By the way, when you use Wandless Magic, the Ministry of Magic usually can't do anything to you because there's no evidence.
When Wagadou students are accused of violating the International Statute of Secrecy, they have a very strong excuse: 'I just waved my hand, I never meant for his jaw to dislocate.'
As for Lord Voldemort killing Old Tom's family and not being discovered, that's because Morfin Gaunt (the uncle) lived in that area, had a criminal record, and readily confessed to his 'crimes.'
(Someone took the blame, and a repeat offender at that, so naturally no one suspected him. Also, Lord Voldemort was already very powerful at the time and might have already possessed the ability to block the Ministry of Magic's trace.)