AFAIK the term "almost surely" means that an event x happens with probability 1, so $\mathbb{P}(x)=1$.
There is also the term "with high probability" which can be read every here and there. But what does it mean exactly? Is there a definition somewhere? Is it probably just the same as "almost surely"? If yes, why does one not say "with maximum probability"?.
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That something happens almost surely means that there are cases that it doesn't happen, but their probability is 0. For example, toss a coin until you get heads. There is a case of you never tossing a heads, namely (T,T,T,...). However, it's probability is zero.
The term is used because it would be wrong to say that every set of coin toss' will result at least one heads. More on the subject, check for example "Probability with martingales" by David Williams.
High probability instead means just that: depending on the thing you're talking about, 0.95 or 0.99 or something like that can be a high probability.
With high probability means probability near to 1, but it is an informal expression.