Large diaphragm capsules usually have diaphragms of about 1 inch in diameter or more (25-28 mm). Small diaphragm capsules, by contrast, have diaphragm diameters of slightly more than ½ inch (12-15 mm). There's a kind of "twilight zone" in between. Diaphragms on many microphones measure about ¾ inch (18-22 mm), and the manufacturers declare those either large or small diaphragms, depending on the kind of microphone they're trying to market. Cosmetics often play an important part. Large diaphragm microphones are usually quite big and the capsule is mounted vertically to be side addressed; small diaphragm microphones are smaller in size, pencil shaped, and end addressed.
It's a common misconception that diaphragm diameter bears some relation to low frequency response. Bass speakers are larger than treble speakers, aren't they? But speakers must produce sound, i.e. move air; microphones are only receivers, they don't move air, the diaphragm is moved. Bass response on small diaphragm condensers, thus, is just as good as on large diaphragm mics. Small diaphragm mics, on the other hand, can have a slightly better response in the highest treble frequencies. But that's mostly theory. Both large and small diaphragm condensers usually have a very wide frequency range that covers all frequencies humans can hear.
Diaphragm size plays an important part in directivity. Large diaphragm mics become more directional at higher frequencies. This means that on-axis sources appear brighter than sound coming from the sides (off-axis). Also, impulse response on large diaphragm mics is not quite as excellent as on small diaphragm mics, because the diaphragm mass is slightly heavier. But that's just a generalisation. Impulse response on LD condensers is still more than sufficient.
Large diaphragm mics, however, do have one advantage over small diaphragm mics that's relevant in many recording situations: Large diaphragm mics have a better signal-to-noise ratio. A larger diaphragm produces a stronger signal while the amplifier noise remains the same. A louder signal above the same noise floor means better signal-to-noise ratio.
Bottom line: small diaphragm mics perform better than large diaphragm mics in many disciplines, but large diaphragm mics have one advantage - low noise - that outweighs other factors in many situations. Also, some theoretical shortcomings may work to your advantage in real life situations. The large diaphragm's uneven directivity over frequency, for instance, is exactly what produces the ever popular "larger-than-life" vocal sound. So, get a small diaphragm mic, if you want to sound natural, and get a large diaphragm condenser, if you wanna sound bigger and better than natural!