As long as everything is positive, there are simple and logical rules for addition and subtracting. These rules can be expressed as addition increase a number and subtraction decreases a number. If you start with a positive number and add or subtract a negative number the rules are exactly the opposite: addition decreases a number and subtraction increases a number. When either number can have any sign the rules are even more complicated. Addition and subtraction are the simplest situations where positive and negative numbers behave differently.
Multiplying or dividing by positive numbers and negative numbers produce distinctly different results. If you look at a number as a line segment that goes from the origin to the number, multiplying by a positive number may change the length, but not the direction. Multiplying by a negative number may change the length, but it definitely will change the direction. Another difference is when working with inequalities. Multiplying or dividing through the inequality with a negative number changes the direction of the inequality -- for example, less than becomes greater than -- but this does not happen when multiplying or dividing through with positive numbers.
Positive integer powers of positive numbers are always positive. Positive integer powers of negative numbers are more complicated. Even powers of negative numbers are positive and odd powers of negative numbers are negative. The zero power of anything -- positive or negative -- is 1. Negative integer powers obey the same sign laws as positive integer powers.
Positive and negative numbers seem most different when you consider roots. The root of a positive number larger than 1 is a smaller positive number. If the positive number is between 0 and 1, the roots are larger than the number but still a positive number. Odd roots of negative numbers behave in much the same manner, but even roots of negative numbers are another story completely.
The problem is that a negative number times a negative number produces a positive number. For a long time it was thought that this meant that you could not take an even root of a negative number, but it turns out that this is not only extremely useful but in some way "completes" the number system. If you allow square roots of negative numbers, all polynomials can be solved. Now we call the square root of -1 "i" -- for imaginary -- and the square root of -3 is 3i. The number system was expanded to accommodate the even roots of negative numbers.