For example, what is the difference between the following two sentences?
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I'm guessing that you learnt 知っています first? This is the polite way of talking about 'knowing'. 知っている is the 'plain form' that you would use when talking to people you know well. In non-past tense these verb forms are those you will find in a dictionary.
いる (plain/dictionary form) --> います (polite form).
Although, now I think about it, if you are comfortable with 会った then maybe you learnt the plain forms first? To make the polite form you change the final vowel sound from う to い and add ます.
かく -> かきます
よむ -> よみます
However if the final sound end in いる or える then the る is simply replaced with ます
たべる -> たべます
いる -> います
Note that there are quite a few exceptions to this latter rule that you will have to learn as you come across them.