Use jQuery/JS to determine the DAY OF WEEK

new Date().getDay();  //0=Sun, 1=Mon, ..., 6=Sat

See also: Javascript Date Object on MDN.

Word of Caution: This returns day of week on the browser. Because the earth is not flat, the day of the week for some users might be different from the day of the week for your server. This may or may not be relevant to your project...

If you are doing a lot of date work, you may want to look into JavaScript date libraries like Datejs or Moment.js

today = new Date()
dayIndex = today.getDay()

... will get you a numeric representation of "today".

0 = Sunday
1 = Monday
2 = Tuesday
3 = Wednesday
4 = Thursday
5 = Friday
6 = Saturday

A clearer and more detailed answer:

var days = [
    'SUN', //Sunday starts at 0
    'MON',
    'TUE',
    'WED',
    'THU',
    'FRI',
    'SAT'
];

d = new Date(); //This returns Wed Apr 02 2014 17:28:55 GMT+0800 (Malay Peninsula Standard Time)
x = d.getDay(); //This returns a number, starting with 0 for Sunday

alert (days[x]);

Working fiddle.

Update

I just found my favorite method while adding the hacktastic bonus at the bottom.

new Date().toLocaleDateString('en', {weekday:'long'})
// returns "Thursday"
new Date().toLocaleDateString('en', {weekday:'short'})
// returns "Thu"
new Date().toLocaleDateString('es', {weekday:'long'})
// returns "jueves"
new Date().toLocaleDateString('es', {weekday:'short'})
// returns "jue."
new Date().toLocaleDateString('fr', {weekday:'long'})
// returns "jeudi"
new Date().toLocaleDateString('fr', {weekday:'short'})
// returns "jeu."

Original Answer

If you only need this once in your page, keep it simple...

["Sunday","Monday","Tuesday","Wednesday","Thursday","Friday","Saturday"][(new Date()).getDay()]

I didn't think it would be that big a deal to extend, but since I have 3 votes for internationalization in the comments...

// dictionary version
({
  es: ["Domingo", "Lunes",  "Martes",  "Miércoles", "Jueves",   "Viernes", "Sábado"],
  en: ["Sunday",  "Monday", "Tuesday", "Wednesday", "Thursday", "Friday",  "Saturday"]
})['en'][(new Date()).getDay()]
// returns "Wednesday"

// list version
langs=['en','es']

[
  ["Domingo", "Lunes",  "Martes",  "Miércoles", "Jueves",   "Viernes", "Sábado"],
  ["Sunday",  "Monday", "Tuesday", "Wednesday", "Thursday", "Friday",  "Saturday"]
][langs.indexOf('es')][(new Date()).getDay()]
// returns "Miércoles"

And finally, the code golf version...

["Sun","Mon","Tues","Wednes","Thurs","Fri","Satur"][(new Date()).getDay()]+"day"

Hacktastic Bonus... If you only care about the abbreviation, you can trim everything else.

new String(new Date()).replace(/ .*/,'')

// or
(''+new Date()).split(' ')[0]

// or
Date().slice(0,3)