Efficient Algorithms
These come in three flavours:

$A$ is a deterministic polynomialtime algorithm if there exists a polynomial ${P}_{A}(\lambda )$ such that for all inputs $X$ of length $\lambda $, ${\mathrm{TIME}}_{A}(X)\le {P}_{A}(\lambda )$.

A probabilistic strict polynomialtime algorithm $A$ has the same definition as above except that $A$ may flip coins.

A probabilistic expected polynomialtime algorithm $A$ has the same definition as above except $E[{\mathrm{TIME}}_{A}(X)]\le {P}_{A}(\lambda )$.
Efficient Interactive Machines
As above, but running bounds should hold for all inputs and all input communication messages.
Negligible Functions
A function $f:{\mathbb{Z}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ is negligible if for all $c>0$, there exists a ${\lambda}_{0}>0$ such that for all $\lambda \ge {\lambda}_{0}$, $f(\lambda )<1/{\lambda}^{c}$.
Example: ${2}^{\lambda},{\lambda}^{\mathrm{log}\lambda}$ are negligible, ${\lambda}^{1000000}\mathrm{is}\mathrm{nonnegligible}$
[cf. Dan's definition: A function $g(t):{\mathbb{Z}}^{+}\to {\mathbb{R}}^{+}$ is negligible if for all polynomials $h\in {\mathbb{Z}}^{+}[x]$ we have that $g(t)<1/h(t)$ for sufficiently large $t$. ]