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Could someone explain what a theory is in the most intuitive way possible? Also, what does it mean for a theory to be maximal? Is this the same as being complete?

Thank you very much in advance for your help.

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In most formal-logical contexts, a theory is simply any set of formulas (sometimes a set of sentences), which you intend to view as axioms -- that is, you're going to be concerned either with either what you can derive from them, or about its models, or possibly both. That intention is the only thing that distinguishes a "theory" from a random set of formulas.

Some authors say that a theory is a logical vocabulary together with such a set of formulas; this is sometimes more intuitive in addition to technically convenient (especially for speaking about models and categoricity), but much of logic can be developed without needing that kind of pedantry, hence the difference between authors.

A "maximal" theory is not quite a standard technical term, but the most sensible technical meaning would be a consistent set of formulas (in a particular language) which has no consistent proper superset. This is a stronger property than being complete; a complete theory merely has to imply one of $\phi$ and $\neg\phi$ for every closed $\phi$, but a maximal one actually has to contain one of them as an axiom.

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    So what is the difference between a theory and a formal system (i.e., a formal language + a deductive apparatus?) – 2017-01-28
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    @Fishermansfriend: A theory is your set of axioms. A formal system is the rules for how you derive things _from_ your chosen axioms. – 2017-01-28
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    If I ask it's because I'm reading some lectures notes by my teacher that say that the terms "theory" and "system" are sometimes used as synonymous. He also writes: «This term (i.e., "theory"), when it is not understood as synonymous of `system', designates a deductively closed set of formulas, that is, a set of formulas such that any formula derivable from the set belongs to the set. A deductively closed set of formulas may not contain formulas that are not provable, in which case it reduces to a set of theorems.» Is this compatible with your explanation? – 2017-01-28