I'll clear this up for you, ships like the Yorktown are United Earth Military ships (say Starfleet ships as I haven't fully decided on the official Earth Military name as of yet).
Now ships like the one posted above and NW classes are first fleet ships. The first fleet was formed from the survivors of a Romulan attack on a number of ships belonging to the Earth Cargo Service. These survivors tired of not being well defended by Earth went to the Earth Council and managed to secure themselves several combat ships, most of older designs and some have been built from the left-overs of several designed ships.
They declared themselves the United Earth First fleet, a Military Elite Group of Earth. However the condition of the handover of these ships was that they would become solely responsible for protecting the shipping lanes of Earth, which they gladly accepted which freed up Starfleet units to head to the front lines and right the war with the Romulans. It was one of the key points in the entire war. They renamed the ships after their old freighter ships or lost loved ones, decided to go for a more traditional Earth prefix system and set out. Over-time they started to develop a real xenophobic opinion towards alien species and they expanded often getting officers whom were rejected from Starfleet Academy or dishonourably dis-charged for a number of reasons The First fleet is still every much active in the early 25th century.
Pretty much they are the official military of Earth, like how the Andorian's still have the Home Guard. While Starfleet became the Federation's united fleet, member planets, especially at the beginning of the Federation reign were still un-trusting of their allies and had their own fleets for protection of their systems. These fleets for the most part were reduced in size as Starfleet became better equipped in the 23rd and 24th centuries however they still remain.
The First Fleet is actually a creation of Dave Metlesits, a friend of Jetfreak's and mine on DeviantArt and isn't a traditional fan, preferring to not be called a Trekkie and has created his own view on the series which I rather like as it makes sense to me.
I hope this clears it up for you now, that is why the Polaris has a Cl registration and the Yorktown and Enterprise classes have prefixes starting with N.
And yes UES stands for United Earth Ship.