The big thing in 2015 are 4K screens. Several years ago, high definition (HD) became the norm. Running at up to 4 times the resolution of the previous standard (SD), it offered sharper, brighter pictures. At this point it is now the norm; aside from the bargain basement no-name sets in the supermarkets, all new televisions are HD and with HD tuners. The resolution on these sets is 1920 x 1080, hence the term '1080p'.
History is now repeating itself - sort of - with 4K television, also referred to as UHD or Ultra High Definition. Once again, the resolution has improved by a factor of four: with screens of 3830 x 2160. Initially these sets sold for silly money, but prices have plummeted in recent months. It will soon be the case that is little to no price overhead associated with 4K, and only the very cheapest sets (back to the supermarkets again) will be 'just' 1080p. So, 4K televisions are posed to just happen regardless.
But even if you can tell the difference, there's not much to watch in 4K. No TV channels. No Blu-ray discs. That will change with time; it's possible to imagine a Sky 4K channel in a year or two, and the first 4K Blu-ray players and a handful of discs should be in the shops by Christmas. Streaming services currently offer the best chance for accessing 4K content, for instance Netflix offers some 4K programming in the form of 'House of Cards'. And House of Cards. And, er, House of Cards (okay, there's a couple of other items but you get the picture. No pun intended). To stream 4K you need an internet connection that will reliably run at 15 Mbits/sec, and even then the pictures have to be heavily compressed. But under the right circumstances it can be very impressive when viewed from 2 or 3 feet. If you like House of Cards.
Guess what is coming next? 8K TV is now on the starting blocks. Once again, resolution goes up by another factor of 4, giving a screen of 7680 x 4320. 8K sets are already available, but are expensive and unlike to come down in price for several years. Japan's NHK (their equivalent to the BBC) are driving much of the move to 8K, but they don't expect it to be mainstream until around 2032. So, no need to worry too much about your new 4K set becoming obsolete overnight.
Although increasing screen resolution is commonly emphasised, it turns out that resolution is not particularly important in our perception of picture quality. Other, more important areas are how rich and vibrant the colours are, plus how bright and dim the screens are. To elaborate... if you look out of the window on a bright sunny day then you may well be dazzled. But a television will not dazzle you - compared to the real world it is actually quite dim and to make the picture more realistic it needs to be a lot brighter. This is harder than it sounds, as the colours still need to be realistic and balanced rather than washed out. Also, bright screens use more electricity, which goes against the grain in these eco-aware times. However, progress is being made and televisions in a couple of years time will be brighter.
At the same time, screens also need to be able to render darker colours more accurately, particularly black. Very few televisions can show true black - it is usually shown as very dark grey, rather than the true absence of light. More expensive televisions have the ability to temporarily switch off parts of the screen's underneath lighting, known as 'local dimming'. However, improvements in screens and standards will considerably improve in this area.
Finally, screens will continue to get bigger and larger sizes will get cheaper. However, anything larger than 55"-60" is not practical in most UK households so this is where the main action will be. Higher screen resolutions demand larger screens to do them justice; it is certainly possible to obtain affordable 40" 4K TVs now but it's rather a waste of time unless you want a rather large computer monitor.
In summary: 4K sets are here now and don't buy anything other unless is a small and/or very cheap television. However, readily available 4K content is some way off plus may never arrive on some platforms, such as Freeview.