Kayak.com does that.
When you do the search, one of the options is to subscribe for updates for the route and dates you've selected, and they'll email you when the prices change.
travel.stackexchange.com/users/1295/littleadv
Skyscanner is another that provides email alerts for price changes, I also use it to find the cheapest flights for any given route as it compares all available flight combinations for you, and if you are flexible about your dates you can view the search in month mode see the price for that route every single day, sometimes you can save significant amount by adding a day on to your arrival and departure days.
Jacks flight club is an email subscription where you can enter your departure airports and they send emails about any cheap flights, limited sales, discounted tickets etc, but I'm not sure if you can specify destination or not.
travel.stackexchange.com/users/90956/Uciebila
Google Flights http://google.com/flights also has price alert option and that is pretty decent.
travel.stackexchange.com/users/72137/Mcload
Yapta.com uses Kayak's alerts for flights and hotels before you travel, but also tracks potential refunds after you buy.
FareCompare.com also has fare alerts for specific city pairs.
However, with the redesign of their site, I'm not sure how to set them up.
travel.stackexchange.com/users/872/arboc7
On http://www.fly4free.com/ you can sign-up for an E-mail newsletter of heavily discounted flights.
Currently its only possible to specify countries you are flying from, and it will send you all deals to any destination from that country.
But its only informing about unusually cheap deals.
travel.stackexchange.com/users/33711/yannn
You could also have a look at the hopper app.
It tries to predict airfare prices
so you know when is a good time to book.
You can find more info here : http://www.hopper.com/corp/announcements/hopper-app-predicts-airfares-and-notifies-you-when-to-book
I use it
and although I don't think it's perfect it does help in giving you a good estimate of a correct, cheap, price.
travel.stackexchange.com/users/50447/DiscoFever
If based in Australia, my site Beat That Flight has cheap flight deals for Australians, focusing mainly on international travel.
It's possible to look at deals based on your state of origin, and get notified when deals come up.
travel.stackexchange.com/users/101/Mark Mayo
I currently spend long time periods outside my hometown and many people advise me to get my ticket for my vacation a month in advance, but I don't want to keep looking each day. Is there an online service that will help me to get cheap flights and allows me to subscribe to it, so I can get email notifications?  If not what do you suggest to get cheap flights from Europe to Central America?
Is there a site where I can subscribe for cheap flights for a specific place?
travel.stackexchange.com
<air-travel><budget><online-resources><tickets>
travel.stackexchange.com/questions/6370
travel.stackexchange.com/users/1563/Necronet
Several companies offer price notifications like this or email alerts of cheap flights, limited sales, discounted tickets etc., including kayak.com, Skyscanner, Jack's Flight Club, Google Flights, Yapta.com, FareCompare.com and Fly4free.com. You could also check the hopper app, or the site Beat That Flight if you are based in Australia. There are a large number of sites you can subscribe to to get email alerts about flight prices and deals, including skyscanner and kayak.
There are a large number of sites you can subscribe to to get email alerts about flight prices and deals, including skyscanner and kayak.