There are 26 leaders in game - every one has his unique combinations of traits, technologies and UU. The traits have the most significant influence on gameplay - other bonuses are rather limited - starting technologies are only important in first 100 turns, UU - has usually rather short time of usefulness (however - it's sometimes can be enough to significantly alter the course of play).
The traits are not equal - some of them are more powerful in early-game (Creative), others are more useful in specific situations (Organized), but most are helpful from start to end. However, all give significant advantages if used correctly.
Free combat I promotion for all melee and gunpowder units
Perfect for warmongers even if this promotion is received only by some of your units (only melee and gunpowder). The combat 1 promotions don't look very powerful - it's only +10% to STR. But you have to remember that is a free promotion - so it means that next promotion will be at 2XP, next 5XP and so on. And what's also very important - combat I gives (after earning 2 XP) instant access to some very useful promotions from the start (like cover, shock, medic) and significantly helps accessing more advanced promotions (march, commando)
Halved costs of Barracks and Drydocks are nice addition that helps more in receiving additional experience.
+2 culture in every city
Priceless in early-game. All your cities quickly expand borders (five turns into first culture radius grown, 40 into next stage) and fast received a defense bonus - without any other investments (monuments, Stonehenge Wonder, religion). This gives a huge advantage especially in first 50 turns, greatly helping in securing resources and land-grabbing. In medium and late phases of the game it becomes much less useful. If you used it well, however, you will have a serious advantage for the rest of game.
Halved costs of Theatre and Colloseum helps in keeping order in cites, and make a possible to build very quickly Theatre in every just build or conquered city - helping in even faster culture development of new cities.
+2 health in all cites.
Nice, especially on higher difficulty levels, when there is less "free" health in all your cities. Helps in making your cities even bigger - works like extra aqueduct.
Halves cost of Harbor and Granary. The former is only useful in coastal cities but the latter is built in every city and helps a lot in city growth.
+1 to commerce on titles already generating at least 2 commerce.
Very powerful trait - every Cottage near rivers, coastal tiles, watermills or windmills is generating extra commerce. Extra commerce means many extra research or extra income. Very nice, indeed.
Halved costs of Bank - even more money.
+50% to Wonder Production
Do you like Wonders? You hate it when someone builds your favorite Wonder before you? Stop this! Take this trait. Useful from start to finish. It stacks with bonuses from strategic resources - making Wonders relatively cheap.
Halves cost of Forge. One of the most useful buildings - extra +25% Production in city available in your cites in halved time.
-50 % civic upkeep
A controversial trait - lot of people consider it broken. In my experience is really useful - but only in specific situations. Ideal for a person who is planning to manage a world spanning empire with a lot of cites.
Civics upkeep is strongly connected with size of civilization here -50% gives you a lot of freedom when it comes to choosing your favorites civics.
Halves cost of Courthouse and Lighthouse. This is even more useful. Now Courthouses are really cheap and can be built as first buildings in cities due to very short construction time. In long-term this can result in some serious savings. Cheap Lighthouses are also helpful - but only in coastal cites.
+100% Great Person points.
Greatly improve amounts of GP points - without addition bonuses doubles it giving you a two times more Great People. If you based your strategy on GP - it is a must have. Possibly the most useful trait of all.
Halved costs of University - very nice, because this is a rather expensive building. Cheaper means you will have much more of them, and that means that your science will be much more advanced.
When changing your civic you will not loose turn on anarchy. Effect - more flexible social system. You like theocracy right away - you have it, you like pacifism - ok, here you are. No anarchy between. Hoverer, it's less powerful than it looks like, due to 5 turn time before you can make any other change - what significantly limits usefulness of this trait.
It's simply a few turns advantage over competitors - turns which are not lost because of anarchy. It can help (sometimes you can win only due to a few turn advantage) - but definitely this trait is not as strong as it looks like.
Halved costs of Temples - much more useful that no anarchy. Temple is one of this few building that can be build in city more than once - every time for every religion. If you based your strategy on religion, that can help a lot. Cheaper Temples mean much more culture, priests, specialists, more happy citizens. More Temples gives opportunity to build even more Cathedrals in medium-phase of the game.