Interfaces most of the functionality of the software 4ti2 available at http://www.4ti2.de/. (The user needs to have 4ti2 installed on his/her machine.)
A d×n integral matrix A (with nonnegative entries) specifies a map from a polynomial ring in d variables to a polynomial ring with n variables by specifying exponents of the variables indexing its columns. For example, if A is a matrix

the map from k[s,t] to k[a,b,c,d] is given by (s,t) ↦ (s^{3},s^{2}t,st^{2},t^{3}).
The toric ideal I_{A} is the kernel of this map. It is minimally generated by the 2minors of the matrix

Given the matrix A, one can compute its lattice basis ideal specified by the integral basis of the lattice A, the toric ideal I_{A}, its Groebner bases, etc. In practice, however, these are nontrivial computational tasks. The software 4ti2 is very efficient in computing these objects.
For more theoretical details (and more generality), see the standard reference: B. Sturmfels, Gröbner bases and convex polytopes. American Mathematical Society, University Lectures Series, No 8, Providence, Rhode Island, 1996.
Note for cygwin users: If a problem occurs during package installation and/or loading, it should be fixed by setting the path inside the file .Macaulay2/initFourTiTwo.m2 to whatever folder 4ti2 is installed. For example, if 4ti2 has been installed in C:/cygwin/4ti2/win32, then the line inside the initFourTiTwo.m2 file will look like this: "path" => "C:/cygwin/4ti2/win32/" . Alternately, the path for 4ti2 may be set when loading the package using the following command: loadPackage("FourTiTwo", Configuration=>"path"=>"C:/cygwin/4ti2/win32/") assuming that 4ti2 has been installed in C:/cygwin/4ti2/win32.
If the package SimpleDoc is not found when installing FourTiTwo.m2, see questions and answers 6, 7, and 8 on the Macaulay2 web site.