It's actually not in the Tory manifesto since not everyone in the party agrees on this and Prime Minister Theresa May used to be against this but supports it now.
To understand why it's a big deal, it's important to understand the pros and cons against it. Wikipedia does have an informative page on that - Expansion of Heathrow Airport.
Basically, businesses and the aviation industry mainly support the idea of building a third runway since it will allow the airport to accommodate more passengers and put the airport on equal footing as other main European airports.
However, people are against it as the construction may cause air pollution and hundreds of homes need to be demolished to make way for the expansion.
This BBC article sums up all the opposition against the expansion.
We need to reduce, not increase, emissions
Heathrow generates 50% of UK aviation emissions. This makes 6% of total emissions, according to Department for Transport figures.
With a third runway and as the UK as a whole cuts emissions, Heathrow's contribution to overall UK emissions would rise significantly by 2050, some calculate to as much as 50% of total.
Health of Londoners at risk
A Greater London Authority (GLA) report suggested Heathrow would even now breach the EU regulations on levels of Nitrous Oxide due to come into force in 2010. Extra flights would only make this worse.
Economic case overstated
Many of the new passengers the extra runway would bring would be transit passengers. According to former BA boss Bob Ayling they would spend little or nothing in London, only boosting airline profits.
Impact on the local area
Transport infrastructure around Heathrow already struggles. The extra demands would create gridlock.
Alternatives
For historical reasons, Heathrow is badly located. It is the only major airport with flight paths over a large capital city, for instance.
Some, like London Mayor Boris Johnson, believe this is a perfect opportunity to start again and propose a new airport in the Thames Estuary.