I am trying to understand this concept and have some difficulties.
For example, can I say that $\alpha$ is the cardinality of $\{1,2,3,...\}=\Bbb N$? And if so, what is $\alpha +1$? I guess it is not the cardinality of $\Bbb N \cup \{\sqrt2\}$ (for example)?
On
In addition to Asaf's response, you asked if $\alpha$ was a cardinal, then what is $\alpha + 1$. Note that cardinal arithmetic is not the same as ordinal arithmetic. So while every cardinal is an ordinal, if you want the successor cardinal of $\alpha$ this is not the same as the successor ordinal of $\alpha$ (as exhibited by the fact that $\omega$ and $\omega+1$ have the same cardinality). The successor cardinal of $\alpha$ is the least ordinal greater than $\alpha$ that is not in bijection with $\alpha$.
Ordinals are not cardinals.
Where cardinals are a notion meant to measure the size of a set in a very raw and structureless sense, ordinals are a notion meant to measure the length of a queue. If you prefer, think of it as the line to the bathroom.
Ordinals refer to a linear order, rather than the cardinality. If $\omega$ denotes the natural numbers with their standard order, $\omega+1$ is the order we have by adding a new element and declaring it larger than all the natural numbers. Then $\omega+2$ would be to add another element on top of that.
Both ordinals $\omega+1$ and $\omega+2$ are countable, that is to say, there is a bijection from each of them to $\omega$, i.e. the natural numbers, but this bijection does not preserve the order itself.