Astraware Mahjong detects if you haven't interacted with the screen (moving around or making a match) for about 10 seconds, and highlights a tile which could be used to make a match - that means one where a partner for it is also visible and uncovered enough to be selected.

Here's an example picture of how the hint highlight looks.  To turn this off (as some people find it distracting!), tap the in-game menu button in the top left of the screen, circled here:

Choose the Settings option from the pop up menu:

On the Game Settings screen, look for the Auto Hints option and tap it to toggle the hints to be on or off.

Does this mean I can't have any more hints?

Even if you choose to disable the auto hints, you can always call up a hint at any time using the small lightbulb+question mark icon in the lower right of the game screen, and it will find a hint for you.