Features:

The HR diagram is a scatter diagram for stars.
Most nature-definingly for stars, it should be a log-log plot of luminosity versus photospheric temperature.
However, neither of these quantities are direct observables, and so proxies are often used as described below.
The HR diagram allows an empirical classification of stars for which a theoretical understanding has developed.
The horizontal axis shown in the diagram is OBAFGKM spectral types:
1. The spectral types run O, B, F, G, K, M with the extension for brown dwarfs L, T.
2. Brown dwarfs are NOT considered stars at all in that they never have significant nuclear burning.
3. The OBAFGKM spectral types are traditionally mnemonicked by "O be a fine girl/guy kiss me."
  Sometimes it's the only sensible thing to say.
4. The stellar spectra of the spectral types and are correlated with photospheric temperature and color index (AKA color) B-V (AKA redness).
5. O to T goes down in photospheric temperature and up in B-V (i.e., up in redness).
6. The spectral types are divided into 10 subtypes (not shown on the diagram).
7. The subtypes run 0,1,2,3,4,5,6,8,9 in order of decreasing photospheric temperature.
8. For example G stars run G0, G1, G2, ... , G9.
9. The Sun is a G2 V star: spectral type G2, luminosity class V.
10. The prominent characteristics of the spectral types are tabulated at, e.g., UCL: The Classification of Stellar Spectra.
The bands on the diagram are the approximate regions of the luminosity classes for stars. They are described in Table: Luminosity Classes below (local link / general link: luminosity_class_table.html):
The shown HR diagram is the common kind showing only the main sequence and the post-main-sequence bands: i.e., the 9 luminosity classes. Other kinds of HR diagram show pre-main-squence star Hayashi tracks (i.e., pre-main-squence star evolutionary tracks) and other evolutionary tracks.
The main sequence (luminosity class V) contains stars doing hydrogen burning in their cores.
The hydrogen burning is nuclear burning, NOT chemical burning. Nuclear burning it is the analogous process in nuclear physics to chemical burning. However, nuclear burning is of order 10**6 more energetic than chemical burning.
The main sequence phase of a star's nuclear burning lifetime is the longest phase, and so most nuclear burning stars are main-sequence stars.
There is a Main-sequence rule given below (local link / general link: star_main_sequence_rule.html):
Giving spectral type and luminosity class pretty uniquely specifies most kinds of star.
So we say full stellar classification consists of giving both spectral type and luminosity class.
The "full" stellar classification by spectral type and luminosity class does NOT completely uniquely specify a star since in both classification schemes the categories are only fairly narrow BINS, and NOT very fine BINS.
Stars are really unique individuals looked at closely.
Also stars of the same "full" stellar classification can have vary from each other in other ways: these variations are are sometimes important.
There are, in fact, other classification schemes to handle some of the these cases (see Wikipedia: stellar classification). But those schemes are beyond our scope.
By the by, there is NO useful way to define an average star since the spectral types vary so widely that any average star you define would overwhelmlingly NOT be like most stars, and so would give NO useful information.
To expand on the preamble, the horizontal axis of OBAFGKM spectral types of an HR diagram can be replaced by one of B-V or, most nature-definingly, photospheric temperature.
In the case of photospheric temperature, the temperature decreases to the right---one of the old wrong-way traditions of astronomy.
When photospheric temperature is the horizontal axis, it is usually in the form of logarithmic photospheric temperature (see Wikipedia: HR Diagram: Forms of diagram).
To expand on the preamble again, the vertical axis shown in the diagram is absolute magnitude in V: i.e., absolute V magnitude.
V band is the passband filter most often used to characterize the brightness of stars.
It has central wavelength λ_central = 0.551 μm and FWHM Δλ = 0.088 μm.
The V band mostly measures in the green color band (∼ 0.495--0.570 μm), yellow color band (∼ 0.570--0.590 μm), orange color band (∼ 0.590--0.620 μm), and red color band (∼ 0.620--0.740 μm).
Absolute V magnitude like all magnitudes is a logarithmic measure of brightness.
Also magnitude runs the wrong way---higher number is dimmer, lower number is brighter---it seemed like a good idea to Ptolemy (c. 90--c. 168 CE)---just accept it.
A change of 5 magnitudes is defined to be exactly a change of 100 in luminosity, (i.e., energy output per unit time). This means that a change of 1 magnitude is a change in luminosity of 100**(1/5) = 10**(2/5) = 2.5118864 ... ≅ 2.512.
The zero point of absolute V magnitude is approximately the absolute V magnitude of stars of luminosity 100 in solar luminosity units L_☉. Recall solar luminosity (L_☉) is the natural unit for luminosity from our human perspective.
Absolute V magnitude for the vertical axis can be replaced most nature-definingly by luminosity (the total energy emitted per unit time by a body) with solar luminosity (L_☉) as the unit.
Actually, there is NOT an exact one-to-one correspondance between absolute V magnitude and luminosity.
Absolute V magnitude measures energy only in the V band whereas luminosity is energy from the whole electromagnetic spectrum.
To go from absolute V magnitude to luminosity requires a bolometric correction which is dependent on the type of star.
However, an approximate fiducial transformation is---as one would expect from the discussion above---that a 1 mag increase corresponds to a decrease in luminosity of a factor of 100**(1/5) = 10**(2/5) = 2.5118864 ... ≅ 2.512 and that a 5 mag increase corresponds to a decrease in luminosity of a factor of 100.