How do you pray correctly in Buddhism? I have been calling myself a Buddhist for as long as I can remember, but I have never done it right. I notice that my mom would chant the mantras from the book, while usually I would just say anything while I pray. I am wondering... is there a right or respectful way of praying in Buddhism? 

Just out of curiosity. Thank you!
is there a right or respectful way of praying in Buddhism?

Praying is conventionally understood as a petition or wish addressed to a deity.

In Buddhism, one might:


  meditate spreading a heart full of love to one direction, and to the second, and to the third, and to the fourth. In the same way above, below, across, everywhere, all around, spread a heart full of love to the whole world—abundant, expansive, limitless, free of enmity and ill will. --SN46.54


These two are similar except in one important way. Praying often involves intention and volition. In other words one quite often prays for a fixed result (e.g., rain, health, etc.). Notice that in the described mettā meditation, there is simply an extension of good will without boundaries, intention or direction. From a Buddhist perspective, intention and volition tend to perpetuate the cycle of suffering from craving particular outcomes. 

This wisdom about intention and volition is also evident in religions with formal prayer. For example, there is the well-known phrase: "be careful what you pray for."

In Buddhism, one cultivates Right View, Right Thought, Right Speech and Right Action in the context of your question. Interestingly, asking about Right View is itself Right View!