What Æ is
Æ (capital) and æ (small) started as a ligature, the letters A and E joined together, but in several languages it is a full letter of the alphabet. Danish and Norwegian alphabets end with Æ, Ø and Å, and Icelandic and Faroese use æ too. In Old English the letter was called ash (æsc), named after the rune for the same sound.
In the International Phonetic Alphabet, æ is the vowel in "cat" as most Americans say it. In English words from Latin and Greek, æ is just a typographic ligature, as in Cæsar or encyclopædia; today these are normally written with a plain "ae" or "e". People also use æ in aesthetic bios, as in æsthetic, for a vintage look.
How to type Æ and æ
On Windows, hold Alt and type 0198 on the number pad for Æ, or 0230 for æ (the shorter Alt + 146 and Alt + 145 also work). On a Mac, press Option + ' (apostrophe) for æ and Option + Shift + ' for Æ. On iPhone and Android, press and hold the A key and slide to æ.
In Microsoft Word, press Ctrl + Shift + &, release, then type a for æ or A for Æ. In HTML, use Æ and æ, and in LaTeX use \AE and \ae.
Æ vs AE, and the Œ ligature
If a name or word in Danish, Norwegian or Icelandic is spelled with æ, keep the æ: it is a different letter, not decoration. When æ is not available, the usual substitute is "ae".
The sister ligature Œ œ joins O and E. It is required in French words like cœur (heart) and sœur (sister). Type it with Alt + 0140 (Œ) and Alt + 0156 (œ) on Windows, or Option + Q and Option + Shift + Q on a Mac. For more special letters, see the letter symbols page, and for Greek letters, the Greek letters page.