Lang vs. lange
The German words lang and lange both relate to the concept of length or duration, but they serve different grammatical functions. The primary difference is that lang is an adjective used to describe nouns, while lange is an adverb used to describe verbs, meaning for a long time.
Lang
A1lang is an adjective that describes the physical length or temporal duration of a noun. As an adjective that comes before a noun, its ending changes to agree with the gender, case, and number of that noun. When it comes after the noun (e.g., Der Tisch ist lang), it does not take an ending.
Der Rock ist zu lang.
(The skirt is too long.)
Er schrieb einen langen Brief an seine Familie.
(He wrote a long letter to his family.)
Das war ein langer und anstrengender Tag.
(That was a long and exhausting day.)
Sie hat wunderschönes langes Haar.
(She has beautiful long hair.)
Ein Meter lang ist nicht genug.
(One meter long is not enough.)
Lange
A1lange is primarily an adverb that means for a long time. It describes the duration of an action (a verb) and, as an adverb, its form does not change. Note that lange can also be the declined form of the adjective lang when it precedes certain nouns, which is a common source of confusion.
Ich habe lange auf dich gewartet.
(I waited for you for a long time.)
Wie lange dauert der Flug nach Berlin?
(How long does the flight to Berlin take?)
Er kann sehr lange die Luft anhalten.
(He can hold his breath for a very long time.)
Wir haben uns lange nicht gesehen.
(We haven't seen each other for a long time.)
Die lange Schlange vor dem Kino war abschreckend.
(The long line in front of the cinema was intimidating.)
Summary
In essence, use lang as the base adjective to describe things (der Tisch ist lang) and lange as the adverb to describe actions, meaning for a long time (er schläft lange). The main point of confusion arises because the adjective lang also becomes lange when it must be declined for certain nouns (e.g., eine lange Pause). The key is to ask what is being described: a noun (adjective lang in its various forms) or a verb (adverb lange).