The root of the evil is that calculators work with a finite number of digits (well, we can't afford machines with infinite resources yet). When you want to represent real-life numbers, some can be huge, some can be tiny, and this raises an issue.
Floating-point:
If the decimal point is fixed, you can' represent them all, there can be so-called overflow or underflow; and for intermediate orders of magnitude, you lose significant digits.
For example, using the $5.5$ fixed-point representation, $127500.$ and $0.0000096695$ cannot be represented, and $\pi=3.14159$ just uses six significant digits, four positions are wasted.
The floating-point representation solves these problems by specifying the position of the decimal point, as in the scientific notation:
$127500.=1.275000000\cdot10^5, 0.0000096695=9.669500000\cdot10^{-6}, \pi=3.141592654\cdot10^0$.
Catastrophic cancellation:
(This concept is actually unrelated to floating-point.)
Assume you want to play with the approximation $\pi\approx355/113=3.14159292\cdots$.
If you want to take the sum with $\pi$, all goes well.
$$\frac{3.14159265+3.14159292}2=3.14159278.$$
But now if you want to take the difference,
$$\frac{3.14159265-3.14159292}2=-0.000000135,$$
only three significant decimals remain. This is a catastrophy, because you started with nine figures and end-up with just three, and this is irreversible ! (Floating-point will not come to the rescue, $-0.000000135=-1.35\cdot10^{-7}$ still has three significant digits.)
This phenomenon makes some problems very difficult to solve numerically.
In some cases, catastrophic cancellation can be avoided. For example, when solving the quadratic equation
$$x^2-1000.001x+1=0,$$
the classical formula says
$$x=\frac{1000.001\pm\sqrt{1000.001^2-4}}2=\frac{100.001\pm999.999}2.$$
Assuming you can only compute with five significant digits, using floating-point, you obtain
$$x=\frac{1000.00\pm999.99}2=999.95\text{ and }0.005,$$ where the second root is very very inaccurate.
You can get a much better estimate by using the fact that the product of the roots is $1$, and the second root evaluates to
$$\frac1{999.95}=0.0010005$$
Best Answer
You're discovering that sometimes the usage of a word in mathematics and related studies is not always as uniform as textbooks often make it out to be.
As the entry in Wiktionary shows, mantissa has at least two very distinct meanings in mathematics. One meaning is the fractional part of a number when that number happens to be used as a base-ten (common) logarithm. Another is an alternative word for significand.
Wolfram MathWorld, as you found, has a definition slightly different from either of these. Their definition of mantissa is the fractional part of a number whether or not that number is being interpreted as a base-ten logarithm.
The definition of mantissa as a synonym for significand is not consistent with the other definitions. If $x$ is a number written in scientific notation as $x = 3.45678 \times 10^2,$ the significand of $x$ is $3.45678,$ but since $x$ written in ordinary notation is $x=345.678,$ the MathWorld definition of mantissa says that the mantissa of $x$ is $0.678.$
When people start writing about IEEE-754 floating-point representation, things get even more muddled. If you take your example of $1.010101001_2 \times 2^6,$ the significand (often called the mantissa in this context) is $1.010101001_2.$ But when it comes time to store this number in IEEE-754 representation, the makers of the standard resorted to a clever trick to squeeze one more bit of precision out of the fixed number of bits of any of their numeric formats. They observed that the most significant bit of a number in binary representation is always $1,$ so it is not necessary actually to store this bit when storing the value of a floating-point number. So the IEEE-754 format only stores the sign bit, some bits that encode the characteristic, and the fractional part of the significand. This format does not store the most significant bit of the significand.
What the Wikihow article is showing you is the part of the significand (or mantissa) that is actually stored in an IEEE-754 floating-point number or a similar format. But it's not actually very much like the MathWorld definition of mantissa, because the fractional part of the significand of a number $x$ (in some floating-point representation) is often quite different from the fractional part of the number $x$ itself.