pouvoir

Inherited from Middle French povoir, from Old French povoir, pooir, from earlier Old French poeir, podeir, from Vulgar Latin potēre, regularization of Latin posse. The /-v-/ was eventually added, after the loss of the intervocalic dental, to break hiatus. The archaic 1SG present indicative puis is from a Vulgar Latin *posseō (rather than possō), for which cf. Old Catalan puix.

Audio:

pouvoir

can, to be able to [ quotations ▼]
Je peux venir ce soir. ― I can come this evening.
may
désastre qui peut nous frapper ― disaster which may strike us
(impersonal, reflexive) to be possible; may, could be
Il se peut que je sois malade.

(literally) It could be that I'm ill / (more naturally) I may be ill. / I could be ill.

Pouvoir is not used with verbs relating to the five senses:
Je te vois ― I (can) see you (not “Je peux te voir”, which is “I can see (i.e. meet) you”)
Regarding the use of pouvoir and savoir, three sub-senses must be distinguished:
- Savoir is used for “know how to do something, have learnt something”.
  Je sais nager. ― I can swim. / I know how to swim. / I've learnt to swim.
- Pouvoir is generally used for “be currently capable of doing something”, but Belgian French uses savoir.
  Je ne peux pas dormir. (France) / Je ne sais pas dormir. (Belgium) ― I can't sleep. / I'm not currently able to fall asleep.
- Pouvoir is always used for “be free, allowed, unhindered to do something”.
  Je ne peux pas sortir. ― I can't go out. / I'm not allowed to go out. / Something prevents me from going out.
Puis (/pɥi/) is an archaic form of the first person present indicative peux. It is still used in inversion or with the conjunction si:
Puis-je vous aider? ― May I help you?