てたまらない is made up of the conjunction particle て, and the negated form of the う-Verb 堪（たま）る meaning 'to bear', 'to endure', or 'to tolerate'. Common translations include 'to not be able to help wanting to (A)', or 'extremely (A)'.
The primary difference between both meanings is that the 'can't help but want to (A)' nuance is seen attached to verbs conjugated with たい, while the 'extremely (A)' nuance is seen conjugated with both い and な-Adjectives. In both of these situations, the more literal interpretation is 'to be unbearably (A)'.

• 来月（らいげつ）公開（こうかい）される映画（えいが）が見（み）たくてたまらない。
  I can't help but want to watch the movie that is going to be released next month.
• 昨日（きのう）は３時間（じかん）しか寝（ね）てないから、眠（ねむ）たくてたまらない。
  I only slept for 3 hours last night, so I'm extremely sleepy.
• 娘（むすめ）をお使（つか）いに行（い）かせたが、ちゃんと一人（ひとり）で行（い）って帰（かえ）ってこれるかが心配（しんぱい）でたまらない。
  I sent my daughter on an errand, but I'm extremely worried about whether she would be able to go and come back by herself.

As always, で will replace て when な-Adjectives are used.