I agree with Bye's answer that praying in the Spirit can mean what is spoken of in Romans 8:26-27 about the Spirit offering up groanings on our behalf when we can't find the words. But there is another passage I think needs to be mentioned as well. Paul himself dealt with those who interpreted praying in the Spirit as meaning speaking in tongues, and he absolutely disagreed with them.
1st Corinthians 14:13-20
13 Wherefore let him that speaketh in an unknown tongue pray that he may interpret.
14 For if I pray in an unknown tongue, my spirit prayeth, but my understanding is unfruitful.
15 What is it then? I will pray with the spirit, and I will pray with the understanding also: I will sing with the spirit, and I will sing with the understanding also.
16 Else when thou shalt bless with the spirit, how shall he that occupieth the room of the unlearned say Amen at thy giving of thanks, seeing he understandeth not what thou sayest?
17 For thou verily givest thanks well, but the other is not edified.
18 I thank my God, I speak with tongues more than ye all:
19 Yet in the church I had rather speak five words with my understanding, that by my voice I might teach others also, than ten thousand words in an unknown tongue.
20 Brethren, be not children in understanding: howbeit in malice be ye children, but in understanding be men.
Praying in an unknown tongue is not praying in the Spirit, and praying in an unknown tongue in public worship doesn't edify anyone, since they don't understand what was said and therefore cannot say "Amen" to the prayer, so Paul says stick with a language everyone can understand in public prayer.
I will add the "the spirit" here seems not to even be the Holy Spirit but the human spirit. The idea behind praying in or with the spirit seems to be to pray with sincerity, i.e. from the heart. Even when it comes to tongues, Paul here attributes it to the human spirit, in verse 14, "For if I pray in an unknown tongue, my spirit prayeth, but my understanding is unfruitful." The important thing in praying "in the spirit" is praying sincerely, from the heart. But when people pray in an unknown tongue, their heart may be in it apart from the intellect, which is what Paul is saying should not be done.