If you google 'best climate in the world', the Canary Islands seem to top many lists. However I'd feel that they're possibly far too windy to qualify.
In the USA, Hawaii and California tend to top most lists.
Europeans seem to find the southernmost countries of the continent the best all year round, with Malta being one that I might nominate myself. (Cyprus is also popular with Europeans for its lovely weather almost all year long although, of course, it's in Asia).
For me though, my top choice might be somewhere on the southern side of Australia. I've been to both Perth and Adelaide in the winter months and found those cities to be very pleasant at that time of the year and, unlike Australian cities further north, the heat generally doesn't become unbearable in the summer months.
Any definition of 'the best weather city' though is bound to be coloured by the reason why one might be there. For example, I've visited Madrid in the height of the summer (when most locals get out of the city and head to the coast if they can) and absolutely loved it as a tourist. However if, say, I was a construction worker in Madrid, having to labour all day under the hot sun, I might well see things very differently!