Polar vs. Amphipathic
The answer is that it is a sliding scale. There is no such thing as "purely polar" or "purely nonpolar." Water is very polar and hydrocarbons are very nonpolar, but there is no clear dividing line between the two categories.

Further confusion in biological molecules such as amino acids can occur, because different individuals categorize them in different ways, considering either just the side chain or the whole molecule. For example the amino acids have the formula $\ce{H2N-CHR-COOH}$ and in the simplest case, Glycine, R=$\ce{H}$. This is clearly a very non polar side chain, but the molecule as a whole is mostly polar, as the side chain is such an insignificantly small part of the molecule. Thus authors who categorize amino acids by side chain (mainly chemists) put Glycine into the non polar category, while authors who categorize amino acids by the whole molecule (mainly biochemists and biologists) put it into the polar category. When it comes to Alanine, R=$\ce{CH3}$ all authors agree it has a non polar side chain and most seem to categorize it as a non polar molecule. 

A simple search of the internet shows that there is very little agreement about the amino acid Tyrosine, R=$\ce{CH2-(C6H4)-OH}$. It is classified as polar, non polar, having a polar side chain and having a non polar side chain by different authors. Clearly it has some characteristics of both, having both a benzene ring and a hydroxyl group. The correct definition would be "moderately polar."  

You might hope that it is possible to give molecules a number which describes their polarity. This can be done to some extent. Googling "solvents by polarity" will give you lists for common solvents. This is, however, complicated if the molecule of interest happens to be acidic or basic. An acidic molecule will be polar in acidic solution but will become charged (and therefore much more polar) in basic solution, due to loss of $\ce{H+}$. Similarly, a basic molecule will be polar in basic solution but will become charged (and therefore much more polar) in acidic solution due to gain of $\ce{H+}$.

An amphipathic or amphiphilic molecule has a part that is highly polar and a part that is highly non polar. These terms also generally mean not just a molecule that is moderately polar, but one in which there is some distance between the polar and non polar parts, so the molecular weight would be at least into the hundreds.

The example you give of a fatty acid is a good one. (The deprotonated, negatively charged fatty acid anion in basic solution would be an even better one.) Another example would be many cell membrane proteins, which have a non polar, hydrophobic part (that stays in the cell membrane) and polar, hydrophilic parts (that extend out into the aqueous media inside and outside the cell.)

In summary, different substances should not be pigeonholed as polar or non polar as it would require that you draw an arbitrary line, but in general the polarity of two different substances can be compared fairly objectively. And pH may make a difference.