Not exactly sure about impl var but I assume it's derived from the options prices to get an avg impl var per strike and than average it to get VIX. @sle can probably explain better.
Historical Vols are just that... past realized volatility of the underling. The difference with IV/VIX as a proxy for IV's (or vice versa)... is that Implied Vols is the expected volatility in the days until maturity, based on the past volatility and future events. Future events have a far higher weight in this because the past doesn't necessarily reflect the future.
That's why basing trading decisions on the difference between Hist Vols / Impl Vols shouldn't be made without looking forward.
So both VIX and IV are forward looking vols.
Convexity is because Varience is basically Volatility^2 (squared).