When we understand something, we perceive the meaning of it. When we fathom something, the perception goes a little deeper. To fathom something is to penetrate to the truth of it, or plumb its depths. The word gets its depth, so to speak, from the noun sense of fathom, a unit of measurement equivalent to six feet used for gauging depth in water. Consequently, one can fathom (measure the depth of) a body of water, or fathom (attempt to get to the bottom of) a difficult or complicated matter. Such depths may seem profound and hard to comprehend, which is perhaps why fathom is so frequently used when full comprehension feels just out of reach.
An idea is a thought or conception, a product of mental activity. A notion is an idea in the most general sense, but the term usually suggests thoughts of more fleeting, vague, or imperfect kind. Whereas an idea can be something fairly worked out or elaborate, perhaps even a plan of action (The student has an idea for how to become an engineer), a notion is more like a whim or impulse—a flurry of thought that hasn't been seriously worked out. Perhaps this half-baked quality explains why notions are so often rejected, challenged, or dismissed! When used in the plural, notions can take on a whole new meaning: small items used in sewing, such as buttons, hooks, and ribbon.
Items or ideas that are similar have a likeness or resemblance, especially in a general way. The adjective akin suggests a closer affinity or shared nature. The primary meaning of akin is "related by blood," with the root word kin meaning "family relatives collectively" or "a relative." Akin doesn't always imply familial connection, but it does suggest a sameness of character or properties, as what might result from shared ancestry. Akin often shows up sentences that compare one experience, sensation, or feeling to another stronger or more intense feeling, as in "What he felt was akin to love."