I'm currently studying Particle Physics and HEP and this acronym is omnipresent. I know it means next-to-leading-order but, what is exactly the physical meaning of LO and NLO?
You say you know about Feynman diagrams, at least at tree level. So you might have seen loop diagrams, i.e. diagrams that contain a loop of internal lines (see e.g. https://en.wikipedia.org/wiki/One-loop_Feynman_diagram).
Now, the whole point of Feynman diagrams is to expand the phyisical quantity we are interested in as a power series, $$\sigma= a_0 + \alpha\cdot a_1 + \alpha^2 \cdot a_2 +\dotsm$$ For this to make sense, the expansion parameter $\alpha$, which usually is (related to) some coupling constant, needs to be small -- for example, in QED, $\alpha\approx 1/137$. Then, the higher-order terms, i.e. those with higher powers of $\alpha$, are suppressed, and the lower-order terms dominate (note that I'm glossing over quite a number of issues here to get the basic picture across).
In other words, the lowest-power of $\alpha$ with a nonzero term $a_i$ gives the most important contribution, called leading order, and the other ones are higher-order. In particular, the next one is the next-to-leading order (NLO), contribution and so on (for example, this paper computes NNNNLO contributions: https://arxiv.org/abs/hep-ph/0610143).
This notation is used in other fields as well: Whenever you have a approximation scheme with a most important term and successively smaller corrections, it makes sense to speak of leading-order, NLO etc.