Several media outlet, mostly UK tabloids and US conservative news aggregators have reported that the US military has raised the US "DEFCON" alert level from 4 to 3 because of tensions with Russia. However there is no evidence that this is so. The last time the DEFCON readiness level was at 3 was during the 9/11 attacks of 2001.
The claim that is being repeated here is simply the opinion of one hobbyist web site, defconwarningsystem.com, run by a small group of people who are interested in prepping for a nuclear war. They like to analyze world events, and they run personal radiation monitoring stations. Their guess of the defcon level is just that, a guess. Probably influenced by their fascination by nuclear war.
They make no claims otherwise, and in fact have a disclaimer
Based on postings on the message board and their contact information, it appears to be administered by "Thomas Lonely Wolf" a writer in Washington State, and Ryan Barenklau, a student at Texas A&M. Neither seem to have any expertise in the topic other than a shared interest in predicting nuclear war.
However this has not prevented crazy news headlines in the British tabloids:
This is not entirely the fault of DWS, who are giving their opinion on the current geopolitical situation. They even make some valid observations about the situation. However they are misleading people by not making it clear from the outset that they are not actually reflecting the official military readiness. Their disclaimers are hidden at the bottom of the page, or in a wall of text. It's very easy for fearmongering sites like the Express to pass is off as something more than it actually is, just the opinion of some guys interested in nuclear war prepping.
[Update Oct 20th, 2016]. As a result of the increase popularity of the site, Ryan Barenklau left to start his own site, meaning DWS is now basically a one-man operation]