The Lazard ring_ is a commutative ring which is
and by Quillen's theorem also
The Lazard ring can be presented as by generators $a_{i j}$ with $i,j \in \mathbb{N}$
and relations as follows
$a_{i j} = a_{j i}$
$a_{10} = a_{01} = 1$; $\forall i \neq 0: a_{i 0} = 0$
the obvious associativity relation
the universal formal group law is the formal power series
in two variables with coefficients in the Lazard ring.
For any ring $S$ with formal group law $g(x,y) \in S[ [x,y] ]$ there is a unique morphism $L \to S$ that sends $\ell$ to $g$.
Lazard's theorem states:
The Lazard ring is isomorphic to a graded polynomial ring
with the variable $t_i$ in degree $2 i$.
(e.g. Lurie lect 2, theorem 4)
By Quillen's theorem on MU the Lazard ring is the cohomology ring of complex cobordism cohomology theory.
Let $M P$ denote the peridodic complex cobordism cohomology theory. Its cohomology ring $M P(*)$ over the point together with its formal group law is naturally isomorphic to the universal Lazard ring with its formal group law $(L,\ell)$.
This can be used to make a cohomology theory out of a formal group law $(R,f(x,y))$. Namely, one can use the classifying map $M P({*}) \to R$ to build the tensor product
for any $n\in\mathbb{Z}$. This construction could however break the left exactness condition. However, $E$ built this way will be left exact if the ring morphism $M P({*}) \to R$ is a flat morphism. This is the Landweber exactness condition (or maybe slightly stronger). See at Landweber exact functor theorem.
for some context see A Survey of Elliptic Cohomology - cohomology theories
Daniel Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. Volume 75, Number 6 (1969), 1293-1298. (Euclid)
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 2 Lazard’s theorem (pdf)