Informally, an inaccessible cardinal number is an uncountable cardinal number which cannot be expressed in terms of smaller cardinal numbers. Consider ℵ0. The set of smaller cardinal numbers is precisely the set of finite numbers. We cannot reach ℵ0 (or higher) by using cardinals like 2k (where k is a finite number) or ∑i=1kai (where k and a1,...,ak are all finite). This makes ℵ0 fundamentally different from what comes below it; it introduces a new concept of size (which we call "infinite").
Formally, a cardinal number a is called inaccessible iff:
This is just a codification of the above.
ℵ0 is explicitly excluded from being inaccessible: we're looking for a new concept of size, and merely being "infinite" is old hat. ℵ1 is not inaccessible, since 2ℵ0≥ℵ1 (it's not a strong limit cardinal; neither is any successor cardinal). ℵω is not inaccessible, since ℵω=ℵ0 + ℵ1 + ... (a sum of ℵ0 smaller cardinal numbers).
So where are the inaccessible cardinals? We don't know if there are any. It is consistent with ZFC that inaccessible cardinals exist, and it is also consistent with ZFC that no inaccessible cardinals exist. Just like ℵ0 requires a special axiom (the axiom of infinity) to exist, so too do other "new concepts of infinity". This may be aesthetically pleasing, but it is hardly mathematically satisfying. Certain ("stronger") other "large cardinal axioms" would imply the existence of inaccessible cardinals, and also give pleasing results (on very small cardinals!); this might be a reason to accept such a large cardinal axiom.
Weakly inaccessible cardinals are (or rather, may be) a slightly weaker concept.