I use Google Pinyin Input. (Which seems to have been deprioritized or something, but still...)
The general shape of input methods that let you produce 汉字 is that you provide some type of input that hints at the character(s) you want, the input method displays a menu of options that match your input, and you select the correct option from the menu. For example, if I'm using pinyin entry and I type `shi`, I can choose from 是, 时, 事, 使, 试, 世, 市, 十, 式, 师, 石, 室, ......, which are all pronounced shi. (And heck, those are just the top 12 suggestions. They mean things like "ten", "be", or "stone". The `shi`s go on for several pages.)
You can enter more than one character at once. If I type `bhys`, I'll see the suggestion 不好意思 ("sorry").
The presented options are chosen based on what the input method predicts I'm most likely to want. They are context-sensitive -- the order of suggestions will change depending on what I typed just beforehand -- and the likelihoods and the phrases are collected from what people elsewhere in the world type. Suggestions can be quite current! Without an internet connection, this would be a much worse experience; the predictions would be wrong or useless much, much more often.