iPhone and apps

So I have an iPhone but not sure how to download apps ? Do I need to connect to my mac? Or can I download straight to my iphone?

Hope someone can help! Being my mac is alittle fried right now and needs some love but need money to give love.....so hoping I can get the most out my iPhone without using my mac

Try using the little icon on your phone that's titled 'App Store'.

I have tried that many times , but always tells me to go to iTunes
I have made an Itunes account. Is that the password I need when
Downloading apps ?????

Yes, you would use your iTunes password to access the iTunes store in order to download any of their music, video or apps.

I have tried this many times , but WHY? Does it tell me to connect to iTunes ?