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This is in the present tense, where you are talking about doing something now: you lie down on the sofa, and you lay down a book.

There are a bunch of ways to remember this part.

How to Remember the Difference
The way I remember is to think of the phrase lay it on me. You’re laying something (it, the direct object) on me. It’s a catchy, dorky, 1970s kind of phrase, so I can remember it and remember that it is correct.

What’s that I hear, music in the background? I know I don’t normally play music, but I love Eric Clapton, and his song Lay Down Sally can actually help you remember the difference between lay and lie… [record screeching sound] because he’s wrong.

To say “lay down Sally” would imply that someone should grab Sally and lay her down. If he wanted Sally to rest in his arms on her own, the correct line would be “lie down Sally.”
It’s pretty easy; you lays something down, and people lie down by themselves.
We don’t have to judge Clapton on his grammar; we can still love his music and at the same time know that it’s grammatically incorrect! In fact, that helps us remember, and we can love him more.

If you’re more of a Bob Dylan fan, you can remember that “Lay Lady Lay” is also wrong. The lyrics should be “Lie lady lie, lie across my big brass bed.”
But again we love BoB Dylan, don’t we???

So How to Conjugate Lay and Lie
Here’s how to conjugate these two verbs:

The past tense of lie is lay, so

Last week, Steve lay down on the floor.
The cat lay in the mud after it rained yesterday.

The past tense of lay is laid, so

Last week, I laid the TPS report on your desk.
Mary forcefully laid her ring on the table.

The past participle of lie is lain, so

Steve has lain on the floor for days.
The cat has lain in the mud for hours.

The past participle of lay is laid, so

I have laid the TPS report on your desk.
Mary has forcefully laid her ring on the table.