What are the alternatives to the Higgs mechanism?
I'm pretty sure that even the brief summarization of all the alternatives will take a book or two. I will try to give a review of basic things from my perspective. Let me from the beginning note that the following classification is not accurate -- different classes may and do overlap.
More scalar doublets (multiplets)
First of all one can introduce more scalar multiplets. Two Higgs doublet model (2HDM) is the most favored, because it is also naturally arises from MSSM. NHDMs are also considered.
Doublets are usually considered, because there is a basic constraint on the quantum numbers of the fields, coming from "rho parameter":
$\rho = \frac{M_W^2}{M_Z^2\cos^2\theta_w}=1$
Which can be satisfied only if $(2T+1)^2-3Y^2=1$ with the most natural solution $Y=1,T=1/2$. Of course there are other solutions, leading to bigger values, but I've never seen anyone seriously considering those.
There is still a lot of freedom to impose some extra discrete symmetries, continuous symmetries, the way these scalars interact with fermions, e.t.c., which leads to many subclasses of such models.
Composite Higgses
The central example is the Little Higgs model where Higgs arrives as a (pseudo-)goldstone boson from some higher global symmetries. Changing the underlying symmetry one obtains the whole class of such models.
Extra gauge symmetries are also considered -- they are usually broken dynamically. Technicolor was the most popular one -- now it not so favored, while I don't think that it was refuted completely. Top condensate is another dynamical model.
Originated from extra dimensions
Lots of geometries, compactifications, boundary conditions -- I feel completely lost with those. Most popular are higgsless models -- attempts to get rid of Higgses completely. They are usually based on some specific boundary conditions.
Of course the list in incomplete. There are a lot of different "mixtures" between those models, usually with some new funny names.
Here is a nice recent reference that reviews some of the mentioned models going more deeply into some tehnical details.