What are the basics of flying non-revenue?
There are a few classes of Non-rev travel.

Jumpseating
Online pass travel
Interline pass travel

Jumpseating
This is open to pilots and flight attendants, air-traffic controllers (not sure if this is currently true, but it used to be), certain company and government officials (e.g. the FAA and others).
In general pilots can only occupy pilot jumpseats and flight attendants can only occupy flight attendant jumpseats.  I know of cases where pilots can sit in the FA jumpseats but this is largely a topic of the Pilots and FA Union contracts.  In general pilots can jumpseat on any other 121 carrier and some 135 carriers.  Note that jumpseating does not necessarily mean you'll be in the cockpit, as current regulations stipulate you should not sit in the cockpit unless there are not seats in the cabin available.
If you list for a jumpseat you will generally be the lowest priority for a cabin seat, boarding after all other non-rev travelers and then in a specified priority order jumpseaters will fill up the cabin and then the cockpit jumpseats until space is exhausted.
Jumpseating is the most versatile of the options, but the most limited in who can use it.  It is good to keep in mind your jumpseat agreements probably allow jumpseating on cargo 121 carriers in addition to passenger 121 carriers, so keep Fedex, UPS, Atlas, etc in mind when travelling.  Your spouse/kids won't be able to join you though.
Online Pass Travel
The specifics of this travel depends on your airline, but I will speak in general terms.  Pass travel is usually available to the employee, the employeers domestic partner or spouse, the employees minor children and the employees parents.  There is usually also a program for a limited number of passes for friends (buddy passes).
Pass travel is usually limited to your airline those you have a business relationship with.  For example a regional carrier can usually pass travel on the legacy carrier they contract with.  A legacy carrier can usually pass travel on its regional contracts.  If a regional contracts with multiple legacies, you sometimes only pass travel on the legacy system of the legacy you perform flights for.  It can be complicated.
There is always a defined priority for pass travel.  In general employees ride before spouses/parents/children who ride before buddy passes (except when travelling together, you may all share the priority of the highest priority pass on the itinerary).  When multiple companies are involved, the priority levels may reflect a higher priority for the company operating the flight employees than the legacy/regional contract carriers.
Again, this will vary greatly from company to company and you will learn the specific rules that apply to you when you are hired somewhere.
Interline travel
These passes are known as ID75, ID90, etc where the number is the percent discount from a full fare ticket.  You can generally buy passes for yourself, your spouse/domestic partner and your children.  No buddy passes.
To get an ID90, you visit your travel office and request an interline ticket for a specific city pair.  For example you can buy an ID90 for Rome - NYC.  You get an old school paper ticket for that route and no specific airline.  When you want to use the ID90, you present it to the ticketing agents for the airline you want to ride that flies the route you want to fly.  They will in turn issue you a non-revue standby boarding pass.  These are generally lower priority than online pass travel when it comes time to board.  These tickets are fully refundable, so they are often purchased as backups to get home when travelling in case you cannot get back with online passes.
How it all works
If you are the employee, you have a company ID and use this to get through security.  If you are the spouse/partner/child/buddy, then you will print out a standby boarding pass and use that to get through security.  You can check bags, which will get tagged standby and should board if you do.  It can be good to remind the gate you have bags checked when you get a seat.
If you are flying on your own carrier or under your legacy brand, you generally have computer access to the passenger boarding totals.  These can often be hard numbers of people booked, checked in and standbys listed.  You can use this to gauge the likelihood of getting on a plane.  If you are travelling offline then a carrier might have a phone number available but they may only tell you &quot;it looks good!&quot; or &quot;it looks bad&quot; and will not divulge numbers.
Once you are at the gate, you play the waiting game.  You will have a boarding priority which may look something like this for a legacy carrier flight:

Legacy pilots
Legacy flight attendants
Legacy retired pilots/FA
The legacies regional contract pilots / FA
Legacy spouse/parents/domestic partner/kids
The regional contract spouse/parents/domestic partner/kids
Codeshare pass agreements
Buddy passes
Interline passes
Cabin jumpseats
Cockpit jumpseats

This will assuredly vary based on who you work for and who they contract with, and is just to illustrate the complexity of the priority structure.  Within each category listed above there is an internal priority that for many is based on date of hire, others based on time of listing.  Again, this varies.
After all the confirmed revenue passengers board, the gate will board the revenue standby passengers, and once they are on and the number of empty seats is known (misconnects, people not in the boarding area 15 minutes prior to departure, etc), they will board according to the non-rev priority.  When the cabin is completely full, no more non-revs may board except for the jumpseaters, who may then fill the available jumpseats.
Putting it together
If you wanted to non-rev from NYC to Miami tomorrow and your company flies that route, then you just wake up in the morning, list yourself, take a look at the boarding totals and head to the airport.  If you don't even want to do that and you are a pilot, you just go down to the airport, find a gate with a plane headed to Miami and list for the jumpseat on that plane.  If the many rules of priority work out, you get on and you go to Miami.
Some considerations
Pass travel often has a dress code.  Don't show up in a tshirt and shorts unless you know you are allowed, you can be denied boarding.  Make sure all of your pass riders are behaving themselves.  This is especially important to impress on anyone you give a buddy pass, as when they screw up as a pass rider, you can have your pass travel suspended as punishment.