First a point that is often mis-stated: The cosmological constant is not the same as dark energy. We observe that the universe is accelerating (growing big at an ever-increasing rate). Since we do not know what is causing this accelerated expansion, we dub the "stuff" that sources this accelerated expansion "dark energy". A cosmological constant (a type a fluid that never dilutes) is one possible explanation for dark energy, but there exist other possible fluids (such as scalar field quintessence) that can also lead to an accelerated expansion.
In gravity (general relativity), go back to the Einstein Field Equations: $G_{\mu\nu}=8\pi G T_{\mu\nu}$. The (energy-density and pressure of the) stuff that fills the universe ($T_{\mu\nu}$) will affect the way the background curves (in other words, $T_{\mu\nu}$ sources gravity).
As the universe expands, the energy density of normal matter will dilute (planets, dust, etc.) proportional to $1/\text{Volume}$. The energy density of relativistic matter will dilute proportional to $1/\text{Volume}^{4/3}$. A cosmological constant will never dilute: the energy density is constant at all times.
The behavior of the background on which all other particles live (e.g. the curvature of the background, or $G_{\mu\nu}$) will depend on the stuff on that lives on background $T_{\mu\nu}$. It is this behavior of the background that is "gravity". A $T_{\mu\nu}$ described by a cosmological constant will grow at an ever increasing rate. A universe that is homogenous and filled with a $T_{\mu\nu}$ that dilutes in a manner similar to normal everyday matter will tend to eventually collapse on itself.
Basically, you may say that the manner in which the energy-density of "stuff" dilutes determines how that "stuff" gravitates.
