There is a whole zoo of topological phases, and hopefully someone will provide a more complete answer, but here are some thoughts.
Symmetry and dimension. The topological classification of a material with a gap (topological insulator or topological superconductor) depends on (i) symmetry and (ii) dimension. These relations are summarized by the so-called tenfold way shown in this table (from this paper):
The "Symmetry" columns correspond to time reversal symmetry (T), particle-hole symmetyry (C), and chiral symmetry (S), with "0" meaning no symmetry, "$\pm1$" is the square of the antiunitary operator of the symmetry. The "Dimension" columns correspond to the spatial dimension (1, 2, and 3 are most relevant for real systems, but some higher-dimensional synthetic systems have also been explored). A well-known example are time-reversal invariant topological insulators (what are typically called topological insulators), which obey time reversal symmetry (and for electrons T$^2=-1$) and fall into class AII. You will see this means that there is no topological classification for 1D, and both 2D and 3D have a $\mathbb{Z}_2$ classification, as is well-known for these materials. Another example are Chern insulators, which fall into class A and admit a $\mathbb{Z}$ classification in even dimensions only.
So how do you figure what are the topological properties of your material? First you need to figure out what the relevant symmetries are and what the dimension is, so that you know where it falls in the tenfold way. For example for a "topological insulator" in 3D, the relevant symmetry is time-reversal symmetry and the dimension is obviously 3. Then, you need to figure out how to calculate the corresponding topological invariant, in this case $\mathbb{Z}_2$. As Shahid Sattar described in their answer, in this case it can be done using a number of standard packages like Z2Pack or WannierTools.
Topological quantum chemistry. Beyond the three symmetries described above, crystalline symmetries further constrain topological order. Additionally, semimetallic systems also admit a topological classification. The database that you refer to in your question uses the formalism known as topological quantum chemistry to classify these phases. In short, it uses the symmetry of the various high-symmetry $\mathbf{k}$ points in the Brillouin zone to determine the degeneracies of the bands at those points, and then uses the symmetries along the paths connecting these points to determine the allowed connectivities of the bands. Each possible connectivity corresponds to a possible topological phase. Then depending on where the Fermi level is, one gets an insulating or a semimetallic phase.
The Topological Quantum Chemistry Database covers almost any material you may ever encounter, so they essentially have done the job for you. However, all their calculations are based on semilocal DFT, and this can be problematic (see for example this paper). Therefore, I would use the database as a good starting point, but then do your own calculations to figure out what is really going on (e.g. using hybrid functionals or the $GW$ approximation to get better estimates fo the bands).
