Upper bound to the number of generators When defining noetherian ring/module there's no condition on the number of generators of ideals/submodules (apart from being finite).
However, in some cases we can do better:
-A noetherian module over a field is a finite vector space, so every submodule can be generated with at most n elements.
-A maximal ideal of $\mathbb{K}[X_1,...,X_n]$ where $\mathbb{K}$ is algebraically closed can be generated, via Nullstellensatz, by exactly n generators.
What other examples are there where we can find a system of generators of bounded cardinality?
What happens if we replace maximal ideal by prime ideal in the second example?
 A: When $(R, \mathfrak{m})$ is a one-dimensional Cohen-Macaulay local ring, then the multiplicity $\mathrm{e}(R)$ is the sharp upper bound for the number of generators $\mu_R(I)$ of ideals $I$ of $R$.  It's sharp because it's also the stable value of $\mu_R(\mathfrak{m}^n)$ for $n$ large (this part doesn't need CM).
A: Understanding the number of generators is a very subtle problem. I will focus on your second question on ideals, since the first one is a bit broad. By a theorem of Foster-Swan, the problem is local.
There is no absolute upper bound even for a prime ideal of height $2$ in $k[[x,y,z]]$. In this paper, Moh gives a sequence of primes $(P_n)$ such that $\mu(P_n) =n+1$.
OK, so what to do next? One can ask if there are good bounds on $\mu(I)$ if $R/I$ is "nice". If $R/I$ is a complete intersection, then $\mu(I)$ is the height of $I$. In commutative algebra, the next level of "niceness" is being Gorenstein.  In this paper Schoutens shows that if $I$ has height $2$ and $R/I$ Gorenstein, then there is a bound only depending  on $R$. 
If one goes down another notch, and only assume $R/I$ is Cohen-Macaulay, then Moh's examples show there are no hope for bounds independent of $I$. However, there are bounds that depends on invariants of $R/I$, such as the type or multiplicity.    
