What are the criteria for a molecule to be chiral?
Mirror images
The very first definition of a chiral molecule is one where it is not superimposable on its mirror image. Therefore, one of the most straightforward ways to determine chirality is to construct the molecule and its mirror image (perhaps via a model), and then to see whether they are superimposable.

Symmetry elements
A fully equivalent definition, rooted in molecular symmetry, is that a species is chiral if it does not have any improper axes of rotation.
An improper rotation is a rotation of the species (say 90° clockwise in the plane of the page), followed by a reflection perpendicular to the axis of rotation (in this case, the page). The improper rotation is generally denoted by $S_{n}$ where $n$ is the number of times you need to rotate to get back to the start, e.g., $n=4$ for a quarter rotation (90°).
So, molecules where you can rotate and reflect them to make them look like the original are not chiral. In other words, it's easier to say when a molecule is achiral. Let's look at some examples of improper rotation that make molecules achiral.
The simplest case to look at is $S_{1}$, but no one calls it that. This is a trivial rotation (a rotation that does nothing) followed by a reflection, or just a reflection, usually denoted by σ. Molecules with planes of symmetry are achiral. This molecule, (2R,4S)-2,4-dichloropentane, is achiral because there is a plane of symmetry that reflects the left side onto the right side.

Another example is $S_{2}$, rotation by 180° then reflection. Generally, we also don't call it that. This is the same as an inversion through the center, which is frequently denoted $i$. Molecules with centers of symmetry are achiral. The following molecule has a center of symmetry. You can also see that rotating it 180° and then reflecting through the page gives the same thing back. For clarity, the mirror image is shown on the right. You can rotate it to see that it's the same.

The above two are the most common symmetry elements that make molecules achiral. Occasionally, you'll run into a higher-order improper rotation axis, like $S_{4}$ in the following example. Rotation by 90° and then reflection through the page gives back the original molecule. The mirror image is shown on the right, and you can rotate it to see that it's the same, thus making this molecule achiral. There are more examples here.

So, looking for the above features will let you know if something is achiral.

Stereogenic elements
To figure out whether or not you even need to consider chirality, you will to find a stereogenic element. Keep in mind that having a stereogenic element doesn't mean that molecule is chiral, because it could have one of the symmetries discussed above and thus, be achiral.
Molecules with stereogenic elements should be considered for chirality, but are not guaranteed to be chiral. In other words, having a stereogenic element is necessary but not sufficient to make a compound chiral.
The main source of chirality that introductory students should worry about is the stereogenic center, usually a carbon with four different substituents:

More generally speaking, a stereogenic center such as this falls into the category of point chirality. Other examples of point chirality include inorganic complexes such as the octahedral $\ce{[Co(en)3]^3+}$ (link, also illustrated below), as well as substituted adamantanes.

Apart from point chirality, here are also two other ways to generate chirality, which are typically seen in more advanced courses.
The first is axial chirality (Wikipedia, IUPAC). This is a twist that can be described as either left-handed or right-handed. These include unsymmetrically substituted allenes, unsymmetrically substituted biphenyls, rings with exocyclic double bonds, and helicenes (image adapted from Wikimedia):

The second is planar chirality (Wikipedia, IUPAC). This is when a plane, which has high symmetry, has its planes of symmetry successively broken by substitution. An example is this borabenzene metal complex. A borabenzene has two planes of symmetry, but these can broken by putting a methyl group on the ring and then binding a metal to the face of the ring.