連用形 (usually translated as "continuative form") is one of "the basic 6 conjugation forms" of Japanese verbs/adjectives. For the ichidan verb 食べる, its 連用形 is 食べ. For godan verbs, many of them have two different 連用形. For example, 書く has two 連用形, namely 書き and 書い.
How can we make a 連用形? Simply, remove ます from the masu-form. For godan verbs, you can create the other 連用形 by removing て/で from the te-form.
Why do the two different forms have the same name? Isn't it confusing? Because they were historically the same. For example, the te-form 書いて was originally 書きて. See this chart for the actual euphonic changes. The term 連用形 is indeed somewhat confusing for learners, and that is probably why you did not learn about 連用形 in your Japanese classes. 連用形 is mainly taught in Japanese classes at Japanese schools.
What does the kanji 用 mean here? This 用 stands for 用言, which is a term that basically refers to Japanese "predicative" words. You can think of it roughly as "Japanese verbs and adjectives".
What are the other names of 連用形? 連用形 is translated variously as "continuative form", "conjunctive form", or "combining form". Specifically, the "masu-form minus masu" version of 連用形 has various alternative names, including "pre-masu form", "masu-stem", "stem" and "i-form". As far as I know, there is no specific name for the "te-form minus te/de" type of 連用形. See: Does "te-form" of a verb always include て/で? Why?
Why do we have to learn 連用形? Beginners may not need this term because they tend to learn 連用形 as part of longer "forms". They learn so-called "te-forms" and "ta-forms" even without knowing they include 連用形. But once you start learning Japanese using monolingual dictionaries, understanding the concept of 連用形 and 助動詞 (auxiliaries) is a must. In addition, many native Japanese speakers post answers here using the term 連用形 because this is how they learned Japanese grammar. Lastly, it may help you understand and memorize long rare "forms".
How is 連用形 used?
- It connects to some auxiliaries and particles including ます, て/で, たい, に and たり/だり.
- Masu-stems can directly connect to another clause/verb, just like te-forms can. For example, 声に出し言う and 声に出して言う mean the same thing ("say it out loud"), although the former sounds stiffer.
- Likewise, a masu-stem appears as part of tons of compound verbs. For example 食べ切る, 走り出す, 受け取る.
- You can form honorific verbs from a masu-stem. (お待ちになる, etc)
- Interestingly, some masu-stems work as a noun.