竭力 and 尽力
jiélì and jìnlì

Both the word and the phrase mean "to do all one can," but 竭力 is an adverb and 尽力 is a verb-object phrase. 竭力 is more of a literary term. When applied to negative or passive things, 竭力 is used more often.

(1)虽然他曾竭力反对这门亲事，但最后他们两个人还是在一起了。
Suī rán tā céng jié lì fǎn duì zhè mén qīn shì, dàn zuì hòu tā men liǎng gè rén hái shì zài yìqǐ le.
Although at first their marriage was opposed, in the end they ended up together.

(2) 他表示自己一定会竭力 (尽力) 帮助她。
Tā biǎo shì zì jǐ yí dìng huì jié lì (jìn lì) bāngzhù tā.
He said he would do all he could to help her.

(3) 他施行改革计划的时候，遭到了保守派的竭力阻挠。
Tā shī xíng gǎi gé jì huà de shí hòu, zāo dào le bǎo shǒu pài de jié lì zǔ náo.
When he implemented his plan he faced strong opposition from the conservative group.

(4) 如果你有自己的理想，就尽力去实现吧。
Rú guǒ nǐ yǒu zì jǐ de lǐ xiǎng, jiù jìn lì qù shí xiàn ba.
If this is what you want, get on with doing it.