My question is: what does 'large' imply? How many digits - is there a
'limit' to the amount of digits for an suitable prime?
A prime $p$ is called large in this case if factoring the multiplication $pq$ with a similar-sized prime $q$ prime is considered infeasible.
Usual sizes of the primes include $2^{1024}$ up until $2^{2048}$. For the recommended size of the modulus (which is roughly twice as long as the primes) refer to keylength.com.
There's no hard limit to the size of the RSA-primes, as there are infinitely many primes and any two will do.
The soft limit is the amount of computation you want to invest to use these parameters. The larger your primes, the longer you'll take to encrypt, decrypt, verify and sign data which is undesirable in many cases. Usually this makes people chose the lowest reasonably secure size (e.g. 1024-2048 bit primes).
In some books they talk about 'bits' and 'strings': I still do not
understand what they mean.
Computers represent numbers as a sequence of bits ($0$ or $1$). The bitlength of a number is said to be the amount of bits ($0$s and $1$s) required to represent said number or equivalently: The bitlength of $n$ is $\lfloor \log_2(n)\rfloor +1$.
A string is generally said to be an arbitrary sequence of bits without any interpretation of its contents (f.ex. as number).
What does distinct mean?
It means what it says: Two primes $p,q$ are said to be distinct if and only if $p\neq q$. The reason for this requirement in RSA is obviously that $n=p^2$ is easy to factor.
