How do you know when to pronounce the with a long vowel and with a short vowel?

You pronounce "the" with a long "e" sound when the following Word starts with a vowel sound, regardless of whether it is spelled with vowel. For example:

the Apple

the hour (in American pronunciation, the h is silent, but may be pronounced in other dialects)

the outset

You pronounce "the" with a schwa sound (very close to a short "u" sound) when the following Word begins with a consonant sound. Examples:

the dog

the man

the uselessness (note that this begins with a consonant "y" sound like the Word "you")

A long "e" is also used when you want to emphasize the Word "the." For example:

He makes THE best Apple pie in the world.

She is THE most accomplished violinist in the history of the orchestra.

I am basing this on standard American pronunciation. These are not hard and fast rules, but rather the way most native speakers talk without realizing it. In fact, my elementary school teachers insisted that we always pronounce "the" with a long e during reading class, but neither the students nor the teachers did it during any other class. If you are not a native speaker, following these rules would help you sound less non-native.