That greatly depends on the size of the uke, whether you play standing up or sitting down, if you’re a boy or a girl, and if you use a strap. To sum up:
If you are sitting down, try balancing the uke in between your legs or on one of your thighs. In fact some of the greatest ukulele players of all time (Roy Smeck, Jake Shimabukuro, the members of the Ukulele Orchestra of Great Britain) rest their ukes on their thigh when they play. Holding it that way will also help with one of the most important aspects of holding the ukulele, i.e., minimising the amount of contact with your body. Please resist the urge to hug your uke (no matter how much you love it/him/her!) because if you hold it tightly you will stop the vibration of the wood, which will in turn deaden the sound.
Resting the ukulele on your thigh is also works when you’re playing larger ukuleles (or both!). Sometimes you will have no choice but to play standing up, though…
If you are standing up, hold the ukulele by two points: (1) between the forearm of your strumming arm and your body, and (2) the underside of your fretting hand. Make sure the thing doesn’t slip out of your hands too often, and you’ll be fine. It might take a bit of practice, but it’s worth it.
Although I personally almost never use one, there are clear benefits to using a strap when standing up. Above all, you will not have to juggle your holding position at all when playing, and for those with large mammalian protuberances, it is kind of a must. When using a strap you can also prance around and even dance, as the lovely virtuoso Taimane Gardner is famous to do.
Now, you can use a guitar or mandolin strap if you have one, but you will have to install a strap button. You can try using the sexy-sounding-but-boring-looking Uke Thong if you feel like it, but I personally would recommend just tying a bootlace around the uke’s body, looped around your neck, and keep going.