If you see an "n" with a "~" above, your browser understands UTF-8 and you can read this page. If you see something else (typically an "A" with a "~" followed by a plus/minus sign) your browser does not understand UTF-8 and you should find yourself a better browser.
Let me show you some examples of what you can do in Unicode:
Here is a Danish word: Blåbærgrød (these are all Latin1 letters).
Here is the French oe ligature: Bœuf.
A Polish l with stroke: Złoty.
Hungarian vowels: Könnyű.
Something bad to say in Turkish: ağır işçi.
The following letters are useful in Esperanto: ĉĝĥĵŝŭ.
This is a famous Greek palindrome: ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ ΜΗ ΜΟΝΑΝ ΟΨΙΝ.
The same with small letters may look like: Νίψον ανομήματα μη μόναν όψιν.
But with Unicode you can do much more than this! For example you can use the Greek alphabet or the Cyrillic alphabet (with which Russian, Bulgarian and many other European languages are written). In the heading above, both the Greek καλώς ορίσατε and the Russian добро пожаловать mean "welcome".
So what about all the non-European languages? Arabic and Hebrew? Chinese? Japanese and Korean? And how about very "exotic" (to me) alphabets like Armenian or Tibetan? Oh yes, Unicode supports all these and many more, too.
And you can use Unicode for Mathematics as well!
The cool thing is that you can write all these characters in the same document. Of course, to see everything correctly it does not suffice to have an up-to-date browser: You must also have all the fonts installed properly on your computer. The font installation procedure depends on what operating system your machine is running (Windows, Unix, etc.).
Character Latin1 Unicode UTF-8 Latin1 code interpr. A0 00 A0 C2 A0  ¡ A1 00 A1 C2 A1 ¡ ¢ A2 00 A2 C2 A2 ¢ £ A3 00 A3 C2 A3 £ ¤ A4 00 A4 C2 A4 ¤ ¥ A5 00 A5 C2 A5 Â¥ ¦ A6 00 A6 C2 A6 ¦ § A7 00 A7 C2 A7 § ¨ A8 00 A8 C2 A8 ¨ © A9 00 A9 C2 A9 © ª AA 00 AA C2 AA ª « AB 00 AB C2 AB « ¬ AC 00 AC C2 AC ¬ AD 00 AD C2 AD  ® AE 00 AE C2 AE ® ¯ AF 00 AF C2 AF ¯ ° B0 00 B0 C2 B0 ° ± B1 00 B1 C2 B1 ± ² B2 00 B2 C2 B2 ² ³ B3 00 B3 C2 B3 ³ ´ B4 00 B4 C2 B4 ´ µ B5 00 B5 C2 B5 µ ¶ B6 00 B6 C2 B6 ¶ · B7 00 B7 C2 B7 · ¸ B8 00 B8 C2 B8 ¸ ¹ B9 00 B9 C2 B9 ¹ º BA 00 BA C2 BA º » BB 00 BB C2 BB » ¼ BC 00 BC C2 BC ¼ ½ BD 00 BD C2 BD ½ ¾ BE 00 BE C2 BE ¾ ¿ BF 00 BF C2 BF ¿ À C0 00 C0 C3 80 Ã[80] Á C1 00 C1 C3 81 Ã[81]  C2 00 C2 C3 82 Ã[82] à C3 00 C3 C3 83 Ã[83] Ä C4 00 C4 C3 84 Ã[84] Å C5 00 C5 C3 85 Ã[85] Æ C6 00 C6 C3 86 Ã[86] Ç C7 00 C7 C3 87 Ã[87] È C8 00 C8 C3 88 Ã[88] É C9 00 C9 C3 89 Ã[89] Ê CA 00 CA C3 8A Ã[8A] Ë CB 00 CB C3 8B Ã[8B] Ì CC 00 CC C3 8C Ã[8C] Í CD 00 CD C3 8D Ã[8D] Î CE 00 CE C3 8E Ã[8E] Ï CF 00 CF C3 8F Ã[8F] Ð D0 00 D0 C3 90 Ã[90] Ñ D1 00 D1 C3 91 Ã[91] Ò D2 00 D2 C3 92 Ã[92] Ó D3 00 D3 C3 93 Ã[93] Ô D4 00 D4 C3 94 Ã[94] Õ D5 00 D5 C3 95 Ã[95] Ö D6 00 D6 C3 96 Ã[96] × D7 00 D7 C3 97 Ã[97] Ø D8 00 D8 C3 98 Ã[98] Ù D9 00 D9 C3 99 Ã[99] Ú DA 00 DA C3 9A Ã[9A] Û DB 00 DB C3 9B Ã[9B] Ü DC 00 DC C3 9C Ã[9C] Ý DD 00 DD C3 9D Ã[9D] Þ DE 00 DE C3 9E Ã[9E] ß DF 00 DF C3 9F Ã[9F] à E0 00 E0 C3 A0 à á E1 00 E1 C3 A1 á â E2 00 E2 C3 A2 â ã E3 00 E3 C3 A3 ã ä E4 00 E4 C3 A4 ä å E5 00 E5 C3 A5 Ã¥ æ E6 00 E6 C3 A6 æ ç E7 00 E7 C3 A7 ç è E8 00 E8 C3 A8 è é E9 00 E9 C3 A9 é ê EA 00 EA C3 AA ê ë EB 00 EB C3 AB ë ì EC 00 EC C3 AC ì í ED 00 ED C3 AD à î EE 00 EE C3 AE î ï EF 00 EF C3 AF ï ð F0 00 F0 C3 B0 ð ñ F1 00 F1 C3 B1 ñ ò F2 00 F2 C3 B2 ò ó F3 00 F3 C3 B3 ó ô F4 00 F4 C3 B4 ô õ F5 00 F5 C3 B5 õ ö F6 00 F6 C3 B6 ö ÷ F7 00 F7 C3 B7 ÷ ø F8 00 F8 C3 B8 ø ù F9 00 F9 C3 B9 ù ú FA 00 FA C3 BA ú û FB 00 FB C3 BB û ü FC 00 FC C3 BC ü ý FD 00 FD C3 BD ý þ FE 00 FE C3 BE þ ÿ FF 00 FF C3 BF ÿThe last column shows what the UTF-8 bit sequence would look like if it was mistakenly interpreted as 8bit encoded iso-8859-1.
You can see another table that shows how UTF-8 works.
Addendum: People who want to use the character € (EURO SIGN), should use UTF-8 (as here), not the lame Windows-1252. The EURO SIGN has number U+20AC which corresponds to the UTF-8 encoding E2 82 AC. In HTML, you can even avoid UTF-8, using the decimal form: €