sensible

The French word sensible may be deceptive to an English reader. It essentially has two meanings: "likely to feel/sense", "capable of being felt/sensed". In other words, it is commonly translated by the English word sensitive (especially in the first meaning), and with words such as noticeable, apparent, tangible, perceptible etc in the second.

It actually covers many meanings of English sensitive, such as "liable to be offended", "liable to have problems", "liable to be affected" etc[1]:

une personne sensible
a sensitive person
cette convention traite d'une matière sensible (EP)
this agreement deals with a sensitive issue
une école en zone sensible
a school in a problem area
il a le coeur sensible
he has a sensitive heart, he's tender-hearted

Here are examples of sensible with the essential meaning of noticeable. Notice how we can find various translations in English, depending on whether the sense is positive or negative, and on the degree of noticeability expressed:

une différence très sensible
a very noticeable difference, a marked difference
une réduction sensible des crédits pour... (EP)
a significant reduction in the budget for...
laisser une trace à peine sensible
to leave a mark that is practically invisible

In some cases where it essentially means "liable to be affected", "liable to take notice", you can also often find more idiomatic translations in English:

"avertissement aux personnes sensibles"
"warning to persons of a nervous disposition"
je suis très sensible à votre intervention (EP)
I have taken your words to heart
il s'est montré sensible à cet argument
he was ultimately convinced by this argument

Especially in technical usage, notice compounds such as:

sensible à la chaleur
heat-sensitive
sensible à la lumière, photosensible
light-sensitive
sensible aux rayons X
X-ray-sensitive

sensiblement, sensibilité

Following suit, the first of these generally means noticeably, markedly, appreciably (e.g. sensiblement différent). The second generally means sensitivity, in its various senses.

In addition, sensiblement is sometimes used to mean "to the naked eye", "approximately". For example:

les chiffres sont sensiblement du même ordre (de grandeur)
the figures are roughly within the same order of magnitude, the figures are "within the same ballpark"

[1] Iin the examples, those marked with "EP" are from translations of sessions of the European Parliament.