Keeping that in mind, phones have their own note taking features. Some have those inbuilt while some phones need additional apps to be able to take down notes. I have seen people jot down stuffs even in their contact list because it is quicker. But I recommend people to have note taking apps on their phone because they are mostly lightweight and their features like being able to make lists, check off lists and synchronization come in handy.
For Android™, a simple search will reveal many free note taking apps in the Play Store. Just try it yourself and search for notes and you will find a list that seems to be never ending to download note taking apps. One might even get confused on which to use. Almost all of them offer the same functionality but here are a few favorites: