What is the difference between R precision and precision at K?
R-Precision

R-Precision is defined as $\frac{r}{R}$, which is the ratio between all the relevant documents retrieved until the rank that equals the number of relevant documents you have in your collection in total ($r$), to the total number of relevant documents in your collection $R$.

Suppose in your collection there are 100 documents in total, 30 of which are relevant ($R = 30$), the rest irrelevant. So you retrieve the first 30 documents (because 30 are relevant in total in your collection) and, say, 10 of them are relevant ($r = 10$). Your R-Precision is then $\frac{10}{30} = \frac{1}{3}$.

Precision@K

Precision@K is defined as the precision at rank $k$, which is more intuitive if you know about the definition of precision itself. Suppose your system retrieves 10 documents, where the relevant ones are at ranks ${1,3,6,7,9}$ and you want to know how 'well' your system does until rank 5 ('well' depends on the objective of your system of course but here we'll say we want high precision at low ranks). So you compute your Precision@5, which is the ratio of the number of relevant documents until (including) rank 5 to the total number of documents until $k$ (this number is $k$ itself). Your Precision@K where $K = 5$ is therefore $\frac{2}{5}$.

Youc an see that R-Precision is a measure that allows for assessment of your entire system whereas Precision@K really only takes the documents until $K$ into account. Depending on what you want to measure, either of them may be suitable.