It's simply where two numbers (or sums, products etc.) are expressed as not being equal to each other - for instance, 6 cannot equal 8, so we say that 6 is less than 8 (6<8), which is an inequality.
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It is a collection of terms which are combined using various mathematical operations such as addition, subtraction, multipplication, division, power, inverse, trigonometric functions etc. It does not have an equality of inequality relationship - which would make it an equation or inequality.
An inequality is when a variable and its coeefecient is greater than something. For example, 5x is greater than 2.
According to the site Math Planet, 'A compound inequality contains at least two inequalities that are separated by either "and" or "or".' In the case of "and", a compound inequality such as x > -1 and x < 2 can also be written as: -1 < x < 2 (I also took this example from Math Planet.) There is no such shortcut for the "or" case.
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sum means the answer of a math problem