What are galaxies?
They're systems of stars, gas, and dust, and something mysterious called 'cold dark matter'.
The stars - more accurately, star systems - are like our own Sun, and brighter all the way to over a million times brighter, bigger, and ~a hundred times more massive; all the way down to a million times dimmer, as small as a city, and ~a hundred times less massive. Some have systems of planets that are far more complex (and interesting) than our own solar system; some have no planets. Check out Planet Hunters if you're interested in finding some of these exo-planets (planets beyond our solar system). The light you see from galaxies comes mostly from the stars in it.
The gas - which includes molecular gas (like CO, carbon monoxide, but mostly just H2), atomic gas (like helium, but mostly just hydrogen), and ionized gas (sometimes called plasma, atoms - and some molecules - which have lost one or more electrons, making them ions) - is mostly so rarified that it's a 'harder' vacuum than any we can create here in our labs, but can be as dense as the upper parts of our own atmosphere (but that's still a pretty darn good vacuum). Some of the light from 'blue' spiral (and irregular) galaxies comes from gas that is being zapped by hot, bright stars (or shocked by supernovae). In some special galaxies, e.g. the Green Peas discovered by zooites a few years' ago, the light from this gas dominates what we see.
The dust is tiny, smaller than the smallest smoke particles or the diesel pollution that so worries people who live in Beijing. It is mostly carbon or silicon ('soot' and 'grit'), often with a coating of water (or other) ice. Some is like really tiny iron shavings (also often coated). We see dust mostly because it makes a galaxy look redder; the more dust, the redder. This is particularly noticeable in edge-on spiral galaxies, which often have a dustlane running down the center.
The 'cold dark matter' is matter of a form we have never seen here on Earth. It is completely invisible, and in total about five times the mass of everything else in a galaxy. We think it is mostly distributed as a dented sphere, kinda like a fishbowl, with the visible galaxy as a small fish in its exact center.
In the very center of almost all galaxies is a super-massive black hole (SMBH), which is exactly what its name is (in mass they range from ~a million times that of our Sun to more than tens of billions). When they are surrounded by (ordinary) matter in a disk (called an accretion disk), they can be incredibly bright (the accretion disk, not the SMBH), even outshining the rest of the galaxy. We see these as AGN (active galactic nuclei), the brightest of which are called quasars.
The stars, gas, and dust in spiral galaxies are arranged in two distinct systems, a disk (a more-or-less flat thing, like ... wait for it ... a disk) and a bulge, which is a more-or-less spherical blob at the center. Within a disk, there may be arms, rings, pseudo-rings, bars, ... In elliptical galaxies there's usually no dust, and often little gas. These are yellow(-ish) blobs, mostly round but sometimes almost cigar-shaped.
So the thing I find hard is classifying edgeon spiral and elliptical galaxies. Any tips?
Formally, the easiest way to distinguish is if the 'axis ratio' is greater than 3:1 (the long axis is more than three times the short one, in length); if so, it's a disk galaxy (spirals are all disk galaxies, but not all disk galaxies are spirals). This comes from extensive studies done on ETGs (early-type galaxies, basically lenticulars and ellipticals; galaxies which look red or yellow): no true ellipticals were found to have a ratio greater than 3:1 (and even those may also be lenticulars - a 'lenticular' is a disk galaxy, usually with a big bulge and no arms).
So, if it looks too cigar-shaped, it's an edge-on spiral. I suspect the 'cigar shaped' classification choice - at the end of 'smooth' tree - is a stand-in for these objects.
Also, if the galaxy is 'pointy', if the ends are not nice and smoothly rounded, then it's a spiral.
Dust lanes are, almost always, a sign of a spiral.
If there's a bar, it's almost always a spiral.