Limited slip and locking differentials
A limited slip (or "posi") differential (or "diff") is a means to apply power to the ground when traction is limited, by sensing which wheel has traction and applying force to it, while maintaining the wheel speeds to be different when turning instead of only to the wheel that is without traction like how an open diff works.
 How to tell an open differential from a limited slip differential (LSD)
Under normal conditions if the rear wheels are lifted off the ground and one of these wheels is rotated, an open diff will allow the rear wheels rotate in opposite directions. A limited slip (or "posi") diff will allow both wheels to rotate in the same direction, and it will be very difficult to hold one wheel from rotating while the other wheel is being turned. But- a damaged open diff or a diff that has been converted by using a spool or mini spool, or a diff that has had the spider gears welded (an old school, dangerous "backyard" method to get two-wheel traction all the time) will ALSO allow both wheels to rotate in the same direction. Conversely, a LSD with a bent housing, worn clutches, or galled pinion gears and/or cross shaft may not allow both wheels to rotate in the same direction with the rear wheels off the ground. Ironically, the way many open diffs get that kind of damage is from extended spinning of the single driven wheel.
Having both rear wheels spin under acceleration is not a positive indication of having a LSD. Under certain circumstances- like when rear wheel traction is identical for both rear tires- an open diff can behave like a LSD.
In some isolated cases like the second generation Pontiac Trans Am, a 10-bolt 8.5" corporate LSD rearend was standard equipment, so they will have a LSD if it's original (except cars specifically ordered with an open rearend).
So, checking whether a differential is limited slip or not isn't as simple as jacking up the rear of the car and turning the wheels, or seeing if both wheels spin when you hammer it. The only way to be 100% certain of what type of differential is in the rearend is to remove the cover or drop the center section and look to see what's inside.
 Other ID cues
Rearend housings will often have codes stamped into the axle tube or center section (depending on make and type of rearend) that can be decoded to give the gear ratio, manufacturer, date of manufacture, LSD or open, etc. Some rearends can be identified as to type by the casting numbers (raised numbers cast into the metal, not stamped), because some rearends use different housings for open and LSD differentials.
Many LSD rearends carried tags saying to use limited slip lube only- a good indicator that the rearend was a LSD originally, but of course the entire rearend or the diff could have been replaced during the life of the vehicle.
 Different ways to get limited slip
Limited slip is accomplished by different means, depending on the manufacturer, application and vehicle use. There is the cone type LSD, there is the clutch type LSD, there are viscous type, there are locking type differentials (mechanical, pneumatic and electric operated) and all accomplish the same basic thing- applying power to the wheel having traction or basically both wheels simultaneously.
Depending on the type and application, there may be some slip allowed, i.e. the two wheels are not connected to one another directly, like in the case of the viscous, clutch, and cone types. In other cases (like the locking-type differentials), the wheels are directly linked while in the locked position.
 "Lunchbox" locker
The lunchbox locker, or "drop-in" locker is a type of automatic locker that replaces the side (axle) gears and spider (pinion) gears (in some/most types) of a diff. The pinion gears referred to here are NOT the same as the pinion gear that meshes with the ring gear. Instead, they're the smaller gears in the center of the diff that allows the two wheels to rotate at different speeds when turning, etc.
A lunchbox locker is relatively easy to install, mainly because the original backlash and preload settings are not disturbed during the installation. The lunchbox locker will only be as strong as the original carrier and rearend housing it's installed in, so using one in a weak design will still be a weak design. One thing that makes it better than some of the less expensive alternatives to a proper LSD is the lunchbox does act as a differential. Not particularly well, though, and because of its idiosyncrasies its generally not recommended for street use because of things like how it can lock while cornering, depending on how the power is applied. More here.
A spool is not a differential, and isn't considered to be limited slip, either. A spool connects both wheels the same as if there were a live axle (think of a go kart from the '60s). In the case of a spool, mini spool, or welded spider gears, both wheels are turning the same speed all the time- even when going around a corner or doing parking lot maneuvers. This causes the inside wheel to rotate faster than it needs to when cornering or maneuvering. The result is reduced tread life, squealing of the tires even at slow speeds when turning, and a lot of stress on the rear end components. Not to mention a potential loss of control in low-traction conditions like ice, snow or rain.
A mini spool is much like a spool, but without the strength of a spool because like the lunchbox locker, it only replaces the spider and axle gears; the rest of the existing rear end is all reused.
 Welded spider gears
A welded diff has the dubious distinction of not only having all the drawbacks of a spool on the street, but also adds the danger of loss of control should the welds break and jam between the ring gear and pinion- which will lock the rear wheels. Not what you want to happen at ANY speed, let alone at the top end of a run, or while carving the canyons.