What kind of changes "stays" in the neuron after the nervous impulse happened?
TL;DR: There are many ways a neuron can change after a neural impulse happens including calcium influx, long term potentiation and depression, and G-protein coupled receptors. While not an exhaustive list by any means I'll give a brief summary of some mechanisms I know best.
Calcium Entry
There exist calcium channels that open in response to a variety of signals be it voltage depolarization (voltage gated calcium channels), neurotransmitter acetylcholine (Nicotinic-Receptor) or Both depolarization and Neurotransmitter binding (NMDA receptor). There is much more calcium outside the cell than inside the cell. Once inside the cell calcium binds to EF-hands in various calcium-binding enzymes like calmodulin. When calcium binds to the EF-hands of the calmolulin protein it activates and from there can interact with the other enzymes responsible for cell signaling and the effects can be quite diverse depending on what the neuron is responsible for doing and the specific signaling cascade it sets off.
LTP/LTD
I'll focus on the NMDA receptor(excitatory Glutamate/Glycine receptor) in the CA1 region in hippocampus. LTP (or Long Term potentiation) occurs when a synapse (connection between neurons) becomes stronger, meaning the post-synaptic neurons response to the same stimulus from the pre-synaptic neuron is stronger after LTP has occurred. LTD (or long term depression) is the opposite effect where a neuron's response to the same stimulus is weaker after LTD.
The LTP whole process is dependent on calcium entering through the NMDA receptor channel. When both glutamate and glycine are released pre-synaptically they open the channel,  allowing calcium ions to flow into the cell. However, If the voltage in the post-synaptic membrane is too low, Magnesium will get stuck in the channel blocking calcium (and the other ions too!). If the neuron has previously been stimulated and depolarized, by another neurotransmitter/receptor system like AMPA the voltage will be positive enough to pop the magnesium out of place and allow calcium into the cell.
Once calcium is in the cell it is bound to calmodulin which then binds to CaMKII (an enzyme activated by the calcium/calmodulin complex). This will cause a long and complicated signaling cascade which results in AMPA (the other excitatory glutamate receptor in the post-synaptic region) to be phosphorylated causing it to be more active. Furthermore the signaling cascade can cause an increase in AMPA receptor creation and subsequent insertion into the post-synaptic membrane. This of course results in LTP.
Many rules in biology one quickly finds exceptions and caveats to nice rules of thumb (In fact I learned of a few just by preparing this answer!) LTD (long term depression) can occur in the hippocampus as well, and is dependent on the influx of calcium! As paradoxical as that seems, there is a negative feed back loop dependent on the levels of calcium ions inside the neuron, such that strong pre-synaptic stimuli create LTP (potentiation) and weaker (lower frequency) stimuli cause LTD (depression) and rare or intermittent stimuli cause no response. (see this and this for reference).
G-Protein coupled Receptors
G-Protein coupled Receptors are more of a general family of proteins than a specific example. They are metabotropic receptors (meaning they do not allow Ion's to flow through them). Rather when bound they release an G-Protein, which like calmodulin binds to other enzymes and activates them allowing them to do their job. They  can respond to many different neurotransmitters (or more generally ligands) and can do lots of different tasks depending on the  on what type of G-Protein it releases.
I'll give the muscarinic(M2) receptor for acetylcholine as an example. This receptor is found in the heart and responds to acetylcholine release from the Vagus nerve. When the GCPR binds to the acetylcholine it releases its G$\beta$ unit and it binds to potassium ion channels in the heart. This causes the outward flow of potassium ions thus lowering the voltage( hyper-polarizing the heart cell).
Furthermore G-coupled proteins can even induce changes in how many are present in the membrane (i.e. they are self regulating). These are produced by internal signaling cascades and are thought to be the reason for tolerance of drugs. Take for example the $\mu$-opioid receptor (activated by morphine, heroine and other similar compounds). These drugs will induce changes in how well the protein response to the morphine and if enough exposure then the cell will actually take the receptor back into the cell body and thus off the membrane in a process called endocytosis. (source here)
Literature reviews worth reading include (all open access):

Calcium signaling
NMDAR meditated LTP/LTD
GPCR's