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Similarly to one-grade conjugating verbs, two-grade conjugating verbs (nidan katsuyō dōshi 二段活用動詞) are also thought to be vowel-ending stem verbs (boin gokan dōshi 母音語幹動詞). The difference between these verbs and one-grade verbs is that two-grade verbs exhibit vowel alternation, thus the word "two" in the name.

There are two classes of two-grade verbs:  upper two-grade conjugating verbs (kami-nidan katsuyō 上二段活用) and lower two-grade conjugating verbs  (shimo-nidan katsuyō 下二段活用). The vowels in their conjugating bases alternate between two vowels. In the case of upper two-grade verbs, the vowels are /i/ and /u/, and in the case of lower two-grade verbs, the vowels are /e/ and /u/.  

Almost all verbs of the upper two-grade conjugating class have since been simplified into the upper one-grade class in Modern Japanese grammar, with the vowel /i/ being the unifying vowel in the stem, but there are also some that became four-grade conjugating verbs (yodan katsuyō dōshi 四段活用動詞) over time. 

The chart below illustrates one example of each kind of upper two-grade verb based on row. Notice how there are some verbs which end in kana that would not be possible in Modern Japanese--づ, ゆ.

①: 恋ふ ultimately became a four-grade verb where it is only used in literary language.
②: 滅ぶ has two possible conjugation forms in Modern Japanese: the four-grade 滅ぶ and the upper one-grade 滅びる. Of the two forms, the latter is deemed to be more correct. 

③: 浴む is an example of sound drift. Looking at its modern upper one-grade form 浴びる, one might assume that its classical form ought to be 浴ぶ, which is in fact another possible form. The reason for this is because the consonant in the stem was closer to /mb/, so writers were simply spelling it with the Kana they thought was closest.  
④: There are only three ヤ行 two-grade verbs: 老ゆ, 悔ゆ, and 報ゆ. Of these, 悔ゆ could alternatively be seen as a ハ行 two-grade verb in the form 悔ふ. 

1. 清見が関を過ぐ。（終止形）
We pass the barrier at Kiyomi.
From the 十六夜.

2. 木の葉の落つるも、まづ落ちて芽ぐむにはあらず。（連体形＋も）（連用形＋て）
Even for falling leaves, it's not that they fall and then bud.
From the 徒然草. 

3. たけき者もつひには滅びぬ。（連用形＋ぬ）
Even the brave will eventually parish.
From the 平家物語.

4. おのづから、このほど過ぎば、見直したまひてむ。（未然形＋ば）
Naturally, if one passes this period, one will no doubt look over it again.
From the 源氏物語.

Grammar Note: ～てむ is the combination of the auxiliary verb ～つ and the auxiliary verb ～む to in this situation to show speculation with confidence.

5. むなしく過ぐる事を惜しむべし。（連体形＋名詞）
You should regret passing your life in vain.
From the 徒然草. 

6. はやく過ぎよ。（命令形）
Pass it quickly!
From the 枕草子.

7. 良頼の兵衛督（ひやふゑのかみ）と申（まう）しし人の家（いへ）を過ぐれば、それ桟敷（さじき）へ渡り給ふなるべし。 （已然形＋ば）
When we passed the home of Hyouenokami of Yoshiyori, he was supposed to have just passed the balcony.
From the 更級日記.

①給ぶ as a lower two-grade verb is the predecessor of the lower one-grade verb 食べる (to eat) in Modern Japanese. In Classical Japanese, it started off as a humble verb before being used in neutral contexts like it is today. It is worth noting, though, that in Classical Japanese, it also had a four-grade form used as an honorific verb meaning "to bestow," being a contraction of 給（たま）ふ. 

②Although all lower two-grade verbs became lower one-grade verbs in Modern Japanese, there is one exception which is arguably still a two-grade verb, and that verb is 得（え）る, the modern form of 得（う）. Its 連体形 is still うる for many speakers, but the one-grade iteration える is acceptable and arguably more widely used.

8. 雨降り日暮る。（終止形）
 The rain fell, and the sun set.
From the 奥の細道.

9. 心かしこき関守（せきもり）侍りと聞こゆ。 （終止形）
They say that there is a clever barrier keeper.
From the 枕草子.

10. 遠（とほ）くの異朝（いてう）を訪（とぶら）へば、 秦（しん）の趙高（てうかう）、漢の王莽（わうまう）、梁（りやう）の周異（しゆい） 、唐（たう）の禄山（ろくざん）、此れ等は皆、旧主（きうしゆ）先皇（せんくわう）の政（まつりごと）にも随（したが）はず、楽しみを極（きは）め、諫（いさ）めをも思ひ入れず、天下の乱（みだ）れんことを悟（さと）らずして、民間の愁（うれ）ふるところを知らざつしかば、久しからずして、亡（ばう）じにし者どもなり。
（とぶらぶ: 未然形＋ば）（極む: 連用形）（入る: 未然形＋ず）（乱る: 未然形＋ん）（愁ふ: 連体形）
If you search abroad to foreign courts,  Zhao Gao of Qin, Wang Mang of Han, Zhu Yi of Liang, Lushan of Tang, they all did not follow their old masters or former emperors, instead taking pleasure to its extreme, paying no mind to admonishments, failing to understand their supremacy was in disorder; having not understand how the public was in distress, their (prosperity) did not last long as they perished. 
From the 平家物語.

11. 都を出でて...日数経（ふ）れば、弥生も半ば過ぎ、春も既に暮れなんとす。　
（出づ：連用形＋て）（経：已然形＋ば）（暮る: 未然形＋なん）
Having left the capital with half of the third month passed upon counting the days, even spring is trying to come to a close.  
From the 平家物語.

12. 朝（あした）に死に、夕（ゆふべ）に生まるる習ひ、ただ水の泡にぞ似たりける。（Passive → 連体形）
Those dying in the morning and those being born in the evening, this way of life is 
Dying in the morning and being born in the evening, this way of life is just like bubbles in water.
From the 方丈記.

13. 近き火などに逃ぐる人は、「しばし」とやいふ。（連体形）
Do people fleeing a nearby fire say, "wait a minute"?
From the 徒然草.

14. 日も疾（と）く暮れよかし。（命令形）
 May the sun set soon!
From the 今昔物語集.

Particle Note: The particle かし creates emphasis and usually follows the 命令形.

15. かくて今日（けふ）暮れぬ。（連用形＋ぬ）
Thus, today came to an end.
From the 土佐日記.