If your tonearm has a standard cartridge mount, you can move the cartridge around under the headshell. This allows you to align the cartridge. You align it at two points, moving it around slightly until you get the best compromise alignment. To do this, you’ll need an alignment protractor. Many turntables include one, and you can also download one for free from vinylengine.com and print it out.
Place the protractor over the spindle as we've done here; if you downloaded and printed your protractor, carefully punch out the center hole and slide it down over the spindle. You’ll see two targets on the protractor, surrounded by lines or grids. Position the tonearm and cartridge over one of the targets. Now gently move the cartridge around until the stylus touches down right on the target and the cartridge body aligns with the lines on the protractor.
Move the tonearm and cartridge to the second target and see if the stylus hits the target and lines up with the protractor grid or lines. If it does, tighten the cartridge bolts slowly, gently, and firmly, being careful not to move the cartridge while you’re tightening. Get the bolts tight, but don’t crank down too hard on them or you could break the headshell or the cartridge.
If the stylus doesn’t align on the second target, try slightly different positions with the cartridge until you find one that gives the best possible alignment on both targets.
Azimuth is the tilt of the cartridge around the tonearm’s central axis. It affects the left/right channel balance. Usually azimuth is adjusted by loosening a setscrew where the tonearm meets the pivot, or where the headshell meets the tonearm.
The easiest way to adjust azimuth is to just eyeball it. Put a record on the turntable, then lower the tonearm gently down to the record. Look at the face of the cartridge and the stylus. Are they standing straight up? If not, lift the tonearm from the record, twist the headshell slightly, and lower the tonearm back down to the record. You can also use a shiny record and look for the reflection of the stylus; if the reflected stylus lines up with the actual one, you’re good.
You can get geeky with azimuth if you want. I set it using test tones from The Ultimate Analogue Test LP and measuring the level of the tones at the left- and right-channel outputs of my phono preamp with my Neutrik NT1 audio analyzer. You could also use an oscilloscope, a voltmeter, or the $250 Fosgometer azimuth meter.