Note that most of this guide is only applicable to low-security systems or to members of corporations that are at war.
SafeSpots: A safespot is a bookmark somewhere deep in space, away from any warpable objects. The easiest way to make one is to make a bookmark quickly while warping between two warpable objects (like from a gate to a dock), or even better is to drop one between two other safespots (so that enemies warping between the warpables don't see you flash by in space, then drop a probe looking to pinpoint you). The safespot is the hiding-place of people being attacked in wartime or of people hiding from pirates in a low-security system. Know that safespots are not nearly foolproof: Covert Ops ships (or anyone using scan probes, for that matter) can drop Scan Probes into space, which will scan the system fairly accurately, allowing their ships to jump straight to you. Therefore it's a good idea to have a few more advanced bookmarks, and if you're in hiding, move between them every minute or so, or constantly if you think someone is after you. To find out if you're being scanned, just unclick the "Use Overview Settings" of your Overview, and scan; scanprobes will then show on your scanner, if there's any in space. Magnetometric scanprobes and the like can scan out ships as well as the standard Spook/Snoop/Fathom probes, but it takes them longer. Note that just because YOU don't see a probe doesn't mean it's not there; if there are parts of the system you can't scan (a distant stargate for example) you should make a safespot near there, because some probe types can scan further than your onboard scanner can detect them.
On to operating the scanner! First off, make sure your Overview is set exactly how you want it. If you don't want half a million Cargo Cans and stations showing up, take them off. You can save those settings as Scanner Settings or whatnot. It make take some twiddling to get right, once the scanner is open.
Open the scanner; it's one of the small round buttons to the left of your capacitor info while you're in flight. Check the box marked Use Overview Settings. Note that you can swivel your ship around in real space, which will change the angle of the scannner beam.
Let's try to search for a cargo can--perhaps marked My Can. Maybe it's a low-security system, and you have forgotten which asteroid belt you left your can in. Since you don't want to fly into any pirates, you'll want to sit somewhere safe (at a safespot or planet, away from the warp-in point) and use the scanner. Note that you may have to move around a bit if it doesn't show up at all, as the scanner's range does not cover entire larger systems.
So: fill the Distance with 9's, and hit Scan (it will revert to the maximum scannable default distance). Start with 360 degrees. Let's say "My Can" shows up at once. So narrow your beam to 180 degrees (this is an option just below the Use Overview Settings tab). If "My Can" is not there scan the other 180. When you find it, narrow it down farther until you've got it to about 30 degrees (or whatever narrows it down to WHERE the can is--if there's only one asteroid belt in that half of the system, you've found the can, but if there's six, you'll have to keep narrowing it down).
Let's say there's three belts all lined up here--one 1,100,345 km out (ten AU is approxiamately 1,400,000,000), one 999,999,999 km out, and one 20,345 km out. You'd now narrow the Distance to 899,999,999 km (so that two belts are in range). If the can is still in range, you can rule out the 999,999,999km belt out, obviously. Then you narrow the scan to 100,000km out. If the can is now gone from the scanner, you know it's not in range--thus farther out than the 20,345 km belt. Then you just run out to the 1,100,345 km belt and pick up your can.
Why is this important? Well, usually it's not, to a miner. But if you DO lose a can in lowsec space, you can hop to a planet and scan for it. More importantly, it's desperately important to traveling through safe systems. Which brings me to: The Reason for Scanning. For safe travel, the scanner is an unbeatable ally. It is the pirate's best friend, finding victims; it is the belt-scouting miner's guardian, telling him there is someone flying a Vagabond from belt to belt in the local .3 space. If you want to know if a gate is camped, dodge to a moon or planet that's in its scan-range and scan the gate you're moving to. Just aim your scanner at it, narrowed to 60 degrees or so, make sure you're in range (if it doesn't list the gate in the scanner but it does on overview, then you're out of range), and check. If you see a bunch of badasses sitting beside it, and you scan a few moments later and they still haven't moved, then pick another path! If the gate does not show up on your overview at all, hold ALT down and look around in space, find a stargate, right-click it and add it to overview.
One final note: When in low-sec space, don't warp to a moon unless you have Control Towers on overview, and have checked in advance, using the scanner, to be sure that none exist at the moon in question. A Control Tower is the base of a POS, or Player Owned Structure, a protected "deathstar" of sorts anchorable at a moon, which will usually (but not always) fire upon any strangers that warp in upon them. Therefore, hopping to a moon in low-sec could easily mean your death. They do not always do this (it depends upon how they're configured by the players who own it) and usually they're there for purposes of mining and/or research rather than simply killing strangers, but this does not make them any less deadly.