If you want to keep videos that you received in a Telegram chat, you just need to download them to your Android phone or tablet.
Steps
Method 1 of 2: Download a single video

Step 1. Open Telegram on your Android
It's the blue icon with a paper plane inside. Usually found on the Home screen or in the application drawer.

Step 2. Select the discussion that contains the video

Step 3. Tap the arrow on the video
It's a blue circle with a white arrow pointing down. The video will be downloaded to the default location where downloads are saved on your Android.
Method 2 of 2: Enable automatic video download

Step 1. Open Telegram on your Android
It's the blue icon with a paper plane inside. Usually found on the Home screen or in the app drawer.

Step 2. Press ☰
This button is located at the top left of the screen.

Step 3. Choose Settings
This option is near the bottom of the menu.

Step 4. Scroll down and tap Data and storage
This command is located under the heading Settings.

Step 5. Select On Wi-Fi Connection
A list of options will appear.

Step 6. Check the box next to Videos
This ensures that your phone or tablet will automatically download the received videos as soon as you are connected to Wi-Fi.

Step 7. Tap Save
The change will take effect immediately.