All matching estimators for the treatment on the treated effect can be written in the form
$$ \frac{1}{n_T} \sum_{i \in \{d_i=1\}} \left[ y_{1i} - \sum_{j \in \{d_j = 0 \}} w_{ij} \cdot y_{0j} \right] ,$$
where $w_{ij}$ is the weight placed on the $j$th untreated observation as a counterfactual for the $i$th treated observation, and $n_T$ is the number of treated persons. The weights satisfy $\sum_j w_{ij}=1$ for all $i$.
Effectively, from each treated observation $i$, you subtract a weighted average of the control observations. Then you take the average of these differences. These weights are specific to observation $i$. Different matching estimators differ in how they construct the weights.
For example, nearest neighbor matching sets the weight to 1 for the single untreated observations closest to $i$ in terms of the propensity score and to 0 for all others. k-NN uses $k$ closest neighbors instead.
Interval matching consists of dividing the range of propensity scores into a fixed number of intervals (which need not be of equal length). An interval-specific estimate is obtained by taking the difference between the mean outcomes of the treated and untreated units in each interval.
Radius/caliper matching takes the mean of the outcomes for untreated units within a fixed radius of each treated unit as the estimated expected counterfactual. You pick the radius.
Kernel matching uses weights that decline with the PS distance. You can think about kernel matching as running a weighted regression for each treated observation using the comparison group data and the regression includes only an intercept term. Here you have to pick the kernel and the bandwidth. Larger bandwidth means further observations will have larger weights.
Local linear matching is very similar, but also included a linear term in PS. Some people will also include higher order polynomial terms.
Finally, you have inverse probability weighting. The basic idea is that you can figure out the expected untreated outcome (in either the treated population or the full population) by reweighting the observed values using the treatment probabilities.
There are some guidelines about how to pick a method here.
There is a list of software and packages that can do matching here. Stata also now has native PSM estimators. In my experience, replicating the output by hand is often very hard once you go past the simplest estimators. However, you can also find examples with output for all of these online, so even if you don't have the software, they will give you a useful benchmark since you can usually track down the data.
