How do you go about giving your game more life and personality?

I'd like to hear your opinions on this.

Have your NPCs say things related to their "daily lives" and relations to other NPCs. Do not go too in depth with it or have them repeat the same thing too often though or your players will think there is a related side-quest and go looking for it.

In my opinion,  having a detailed world, with NPCs who have their own lives, like Cadh2000 says, can help a lot.  For example, compare Rune Factory (the original) to Rune Factory 3.  In the first one, NPCs basically stood around until their "shift" was over, then they walked to their next spot. 

By the third one, the NPCs were sometimes running from house to house, carrying something or other.  And if you talked to them, they said something related to what they were carrying.  They would repeat, but it definitely added to the feel of a living town.

Also, having little festivals or random events, like a visiting dancer, can help.  The Harvest Moon (and Rune Factory) series use the festivals as a mainstay of the town's life. 

With a sprawling world, you can add pilgrims going to visit a holy site, or other non-hostile random encounters which add to the flavor of a place.  Maybe, every once in awhile, there is the King's Bounty, where the king shares his munificence with his subjects, which is a very large festival that draws the entire kingdom.

Adding small side quests can also help.  Maybe it doesn't matter to the Main Plot that the little girl lost her doll, and you just get a kiss on the cheek in thanks, but it adds to the feel of people who have their own concerns apart from the larger ones driving the player.

Also, adding relevant post quest updates to NPCs can help.  Say you defeat the Drunken Dragon in the beginning, but if you go back late in the game, the Dragon is now friends with the chimney sweep, and you get a small keepsake.  Or, after you defeat the Head of the Thieves Guild, some Rogues are now jostling for the leadership and send you on minor side quests.

Memorable characters (personality, graphics...), nice atmosphere (mapping, weather, HUD...) and fluid and intricate plot (basically not to bore and keep the player interested in the story; if things need to go slow just add humor).

Also originality.

"The Legend of Zelda : Majora's Mask", "The Elder Scrolls IV : Oblivion" and "The Elder Scrolls V : Skyrim" all have one thing in common, their npc's have a daily agenda, consist of unique dialogue, and the majority of non player characters have a side quests. To give your game or project a rich personality, you should program or set up your npc's so that the player can interact with them in different ways. For example, an npc could be performing an action based on the time of day such as digging in a mine shaft, sweeping the floors of a shop, painting a portrait of another npc which can then turn into a future quest for the Player to explore. Another way to give your game a rich personality is to make it non-linear with multiple endings; this gives the Player the option to explore more content in the game which in turn delivers more of a story line.

Have characters talk in their own way.

Let's look at a fantastic example!

Mass Effect 1.  It has loads and loads of Lore and conversations you can have with crewmates and other random people you meet.  What's even better?  If you're even MORE curious about the world, there's the "Codex".  The codex has written entries about the world, its science, how things work, how things came to be the way they are, and even stuff about the various races.  You know what else is cool about the Codex?  Half the entries are voiced!

If you want your world to feel more alive and interesting, just flesh it out.  How did the bad guy rise to power?  Why do people follow him?  How are towns sustaining themselves?  Is there trade?  What about backstories for NPCs?  Everyone is the hero of their own story, so what do NPCs think makes them the heroes of their stories?  Why are towns set up where they are?  How did the castle in the mountains get built?  Why does magic work?  Who makes all the legendary weapons?

Flesh out absolutely everything.  Some of it you won't use in the game, but if you flesh it out, then your players will know it's fleshed out by the dialogue and such that they are given.